BIREP – Representations of finite dimensional algebras at Bielefeld

• BIREP
• Faculty of Mathematics
• Bielefeld University

 

Seminar

No talks have been announced for this week.

For a regular email announcement please contact birep.

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Future Talks

Friday, 14 April 2023

• 13:15
  Alexander Pütz (Bochum): tba

Friday, 23 June 2023

• 13:15
  Sondre Kvamme (Trondheim): tba
• 14:30
  Karin Erdmann (Oxford): tba

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Seminar Archive

Grzegorz Bobinski made his notes from some of the seminar talks available on his web page.

Friday, 20 January 2023

• 13:15, Room V5-148
  Markus Schmidmeier (Boca Raton): Hammocks to visualize the support of finitely presented functors
  Abstract: Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered as a module over a subalgebra.
  When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor.
  Examples include quiver representations and invariant subspaces of nilpotent linear operators.

Friday, 16 December 2022

• 13:00, Room V5-148
  Daniel Bissinger (Kiel): Kronecker representations and Steiner bundles
  Abstract: Let d < r be natural numbers, K_r be the generalized Kronecker algebra with arrow space A_r and Gr_d(A_r) be the Grassmannian of d-planes.
  Jardim and Prata have shown that the category of Steiner bundles on Gr_d(A_r) is equivalent to a full subcategory of mod K_r.
  We identify the objects of this category as relative projective Kronecker representations and give a homological description of the subcategory.
  Then we explain by means of examples how questions regarding bundles can be answered in mod K_r and vice versa.
  This talk is based on joint work with Rolf Farnsteiner.

Friday, 01 July 2022

• 13:15, Zoom
  Haruhisa Enomoto (Osaka): From the lattice of torsion classes to wide and ICE-closed subcategories
  Abstract: For a module category, we can consider the posets of various subcategories. In this talk, I explain that we can compute the posets of wide and ICE-closed (Image, Cokernel, Extension-closed) subcategories from the lattice of torsion classes in a purely combinatorial way. Also I give a simple expression of the poset of wide subcategories from the lattice of torsion classes, and give an application to the combinatorics of Reading's shard intersection order on Coxeter groups.
  

Friday, 24 June 2022

• 13:15, Room T2-149
  Vanessa Miemietz (Norwich): Uniqueness of bocses corresponding to a quasi-hereditary algebra
  Abstract: Koenig-Kuelshammer-Ovsienko proved that an algebra is quasi-hereditary if and only if it is Morita equivalent to the right algebra of a normal directed bocs. I will review their construction and explain my joint work with Julian Kuelshammer, which proves that the basic bocs associated to a Morita equivalence class of quasi-hereditary algebras is unique up to isomorphism.

Friday, 17 June 2022

• 13:15, Room T2-149
  Georges Neaime (Bielefeld): Towards the linearity of complex braid groups
  Abstract: Complex braid groups are a generalisation of Artin–Tits groups. They are attached to complex reflection groups, which are themselves a generalisation of finite Coxeter groups. It is an ongoing challenge to extend the theory of Artin–Tits groups to all complex braid groups. A part of this theory extension was established by Broué, Malle, and Rouquier in their seminal work. An important feature of spherical Artin groups is that they are linear groups, i.e., they admit a faithful linear representation of finite dimension. For the usual braid group, this property was shown to hold independently by Bigelow and Krammer. We seek to extend the theory of linearity to the context of complex braid groups, with a focus on the infinite families. Indeed, we will describe a definition of BMW and Brauer algebras, from which we can construct suitable linear representations and conjecture their faithfulness. We present a number of theorems and conjectures related to the structure of the aforementioned algebras, as well as properties of the relevant representations. We will finally propose a research programme for the sequel.

Friday, 20 May 2022

• 13:15, Room T2-149
  Estanislao Herscovich (Grenoble): Vertex algebras and 2-monoidal categories
  Abstract: In an attempt to make the theory of vertex algebras more natural, R. Borcherds proposed a new foundation based on two tensor products. In this talk I will explain how these ideas sit well within the framework of 2-monoidal categories. More precisely, I will present a certain 2-monoidal categories of functors and show how vertex algebras can be regarded as commutative algebras with respect to one of the tensor products.
• 14:30, Room T2-149
  Thorsten Heidersdorf (Bonn): Categorical quotient constructions and representations of the general linear supergroup GL(m|n)
  Abstract: Finite dimensional representations of complex algebraic supergroups are nowadays reasonably well understood as abelian categories (in terms of blocks, extensions etc) but their monoidal structure is largely unknown. I will describe some quotient constructions to approximate this monoidal structure in the case of GL(m|n).

Friday, 13 May 2022

• 13:15, Room T2-149
  Isaac Bird (Prague): Duality and definable categories in triangulated categories
  Abstract: Over a ring, a submodule is pure if and only if its dual is a summand, while a module is pure-injective if and only if it is a direct summand of a dual module. I will discuss how these statements can be generalised to triangulated categories through the framework of duality triples, before turning attention to duality and definable categories. This is done through the use of duality pairs, which I will introduce. Several applications and examples will be given. This talk is based on joint work with Jordan Williamson.

Friday, 22 April 2022

• 13:15, Room T2-149
  Achim Krause (Münster): Integral endotrivial modules
  Abstract: Endotrivial modules are invertible objects in stable module categories, i.e. modules "invertible modulo projectives". These have been studied intensively over p-groups with coefficients in a finite field, and more recently also for more general finite groups. In joint work with Jesper Grodal, we generalize stable module categories to arbitrary coefficients, and study the resulting notion of endotrivial modules over arbitrary rings, with special focus on integer coefficients. The relationship between the integer coefficient case and the finite field coefficient case leads to surprising character-theoretic integrality conditions.

Saturday, 09 April 2022

• Workshop "Geometry and Representations"

Friday, 04 February 2022

• 13:15, Room T2-205
  Jun Maillard (Saint-Etienne): A Cartan-Eilenberg formula for Mackey 2-functors
  Abstract: The Cartan-Eilenberg stable elements formula expresses the (mod p) cohomology of a group G as a limit of the cohomology of its p-subgroup. This can be generalized by replacing the (mod p) cohomology functor by any global cohomological Mackey functor. A large class of 'categories indexed by finite groups', such as the categories of modules and their stable and derived versions, can be endowed with a structure of Mackey 2-functors, a 2-dimensional counterpart of Mackey functors. I present an analogous Cartan Eilenberg formula for these Mackey 2-functors, and how it can be reformulated as a structure of (higher) sheaf.

Friday, 28 January 2022

• 13:15, Zoom
  Charley Cummings (Bristol): The left-right symmetry of the finitistic dimension
  Abstract: The finitistic dimension is a numerical invariant associated to a finite dimensional algebra that is a measure of the complexity of its representation theory. This dimension can be defined in terms of left or right modules. In general, the left and right finitistic dimensions of an algebra are not equal. However, it is unknown if the finiteness of these two dimensions is connected. In this talk, we consider the connection between the finiteness of the left and right finitistic dimensions of an algebra and relate this to the longstanding, open finitistic dimension conjecture.

Friday, 21 January 2022

• 13:15, Room T2-205
  Wassilij Gnedin (Bochum): Lifting derived equivalences and Serre duality for gentle Gorenstein algebras
  Abstract: Recent work by Iyengar and Krause established Serre duality results for a notion of Gorenstein algebras which include symmetric orders as well as gentle algebras. My talk is concerned with the derived representation theory of these algebras.
  In the first part, I will discuss whether the tilting property of a perfect complex over a Gorenstein algebra is local. This turns out to be true for symmetric orders in a broad sense that allows to consider other rings than localizations at prime ideals. After base change to a complete local ring R, the derived equivalence problem of a symmetric R-order can be reduced to a finite-dimensional setup.
  The second part deals with an explicit description of the dualising bimodule of certain infinite-dimensional gentle algebras, which allows for a homological point of view on their string and band complexes. Combined with Auslander-Reiten theory of orders, this description yields an approach with modest use of combinatorics to recover results on the singularity category of a gentle Gorenstein algebra.
  This talk is a report on joint work with S. Iyengar and H. Krause.
• 14:30, Room T2-205
  Didrik Fosse (Trondheim): A combinatorial rule for tilting mutation
  Abstract: Tilting mutation is a way of producing new tilting complexes from old ones by replacing one indecomposable summand. In this talk, we give a purely combinatorial procedure for performing tilting mutation of suitable algebras. For the path algebra of a (sufficiently nice) quiver with relations, this procedure allows us to perform tilting mutation of the algebra by only modifying the quiver with relations.
  As an application, we show how the mutation rules can be used to show that two algebras are derived equivalent.

Friday, 14 January 2022

• 13:15, Zoom
  Shiquan Ruan (Xiamen): Nakayama algebras and Fuchsian singularities
  Abstract: In this talk, we will classify all the Nakayama algebras having Fuchsian type, that is, derived equivalent to the extended canonical algebras, or equivalently, derived equivalent to a kind of stable category of vector bundles over weighted projective lines. This is achieved by constructing certain tilting complexes in the bounded derived category of coherent sheaves and also in the stable category of vector bundles for weighted projective lines, and the strategy of one-point extension and the perpendicular approach will be used. As a byproduct, we reprove the classification result of Nakayama algebras of piecewise hereditary type due to Happel—Seidel.
  This is joint work with Helmut Lenzing and Hagen Meltzer.

Wednesday, 08 December 2021

• 10:15, Room V4-112
  Hernán Giraldo (Medellin): Shapes of Auslander-Reiten triangles in the stable category of modules over repetitive algebras
  Abstract: Let k be an algebraically closed field, let A be a finite dimensional k-algebra, and let  be the repetitive algebra of A. For the stable category stmod(Â) of finitely generated left Â-modules, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in stmod(Â). We use the fact (and prove) that every Auslander-Reiten triangle in stmod(Â) is induced from an Auslander-Reiten sequence of finitely generated left Â-modules. Finally, we will talk about applications of this last result.

Friday, 03 December 2021

• 13:15, Room T2-149
  Tiago Cruz (Stuttgart): Relative dominant dimension and quality of split quasi-hereditary covers
  Abstract: The dominant dimension of a finite-dimensional algebra A is a homological invariant measuring the connection between module categories A-mod and B-mod, where B is the endomorphism algebra of a faithful projective-injective A-module.
  In this talk, we will discuss generalisations of dominant dimension and how they can be used to measure the quality of split quasi-hereditary covers. The Schur algebra S(d, d) together with its faithful projective-injective module is a classic example of a split quasi-hereditary cover of the group algebra of the symmetric group on d letters.

Friday, 26 November 2021

• 13:15, Room T2-149
  Tobias Barthel (Bonn): Stratifying integral representations of finite groups
  Abstract: Classifying all integral representations of finite groups up to isomorphism is essentially impossible. In this talk, we will introduce an integral version of the stable module category for a finite group G and then explain how to use it to give a `generic' classification of integral G-representations. Our results globalize the modular case established by Benson, Iyengar, and Krause and relies on the notion of stratification in tensor triangular geometry developed in joint work with Heard and Sanders. Time permitting, I will discuss some further directions.

Friday, 19 November 2021

• 13:15, Room T2-149
  Adam-Christiaan van Roosmalen (Hasselt): Auslander's formula and correspondence for exact categories
  Abstract: Following Auslander's philosophy, one can study a small abelian category A by studying the category of finitely presented functors on A. Auslander's formula provides a way to move back: one can recover A as the quotient of the category of finitely presented functors on A by the Serre subcategory of all effaceable functors.
  One impediment to formulating a version of Auslander's formula for an exact category E is that the category of finitely presented functors on E is independent of the chosen exact structure: it only depends on the underlying additive category. To address this, we introduce the category of admissibly presented functors on an exact category. In this talk, I will focus on this category of admissibly presented functors, and use it to formulate a version of Auslander's formula and correspondence for exact categories.
  This talk is based on joint work with Ruben Henrard and Sondre Kvamme.

Saturday, 13 November 2021

• Workshop "Representations of Algebras and Sheaves"

Friday, 12 November 2021

• 13:15, Room T2-149
  Scott Balchin (Bonn): The smashing spectrum of a tt category
  Abstract: In joint work with Greg Stevenson, we prove that the frame of smashing tensor ideals of a big tt-category is always spatial. As such, by Stone duality, we are afforded a space: the smashing spectrum. In this talk, I will report on the construction of this new invariant via lattice theoretic techniques, and its relation to the Balmer spectrum. In particular, we will see that there is a surjective comparison map which detects the failure of the telescope conjecture.

Friday, 29 October 2021

• 13:15, Room T2-149
  Håvard Terland (Trondheim): Identifying components of mutation quivers
  Abstract: Tau-tilting theory completes tilting theory from the perspective of mutation. Letting points be support-tau tilting pairs and arrows indicate (left) mutation, one then obtains a so-called mutation quiver whose underlying graph is regular.
  The goal of this talk will be to introduce tau-tilting theory and discuss recent efforts to better understand the connected components of (the underlying graphs of) mutation quivers of support tau-tilting pairs.

Friday, 22 October 2021

• 13:15, Zoom
  Kaveh Mousavand (Kingston): Orbits of bricks of finite dimensional algebras
  Abstract: In 2014, Adachi-Iyama-Reiten introduced the tau-tilting theory of finite dimensional algebras to fix the deficiency of mutation of tilting modules. The subject soon received a lot of attention and developed in various directions. Around the same time, Chindris-Kinser-Weyman studied the moduli spaces of quiver representations, in particular the behaviour of Schur representations (bricks) of finite dimensional algebras. In this talk, we try to relate these two lines of research. More specifically, motivated by a conjectural geometric counterpart for the algebraic notion of tau-tilting finiteness, we treat module varieties of finite dimensional algebras, in particular the orbit of bricks. We show that, unlike arbitrary bricks, those used for the labelling of the lattice of functorially finite torsion classes always admit open orbits. From this, we obtain a conceptual proof of the counterpart of the first Brauer-Thrall conjecture for bricks. We push these results further in the treatment of another Brauer-Thrall type conjecture which is still open in full generality.
  This is based on my Joint work with Charles Paquette (Royal Military College, Canada).

Friday, 10 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Thursday, 09 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Wednesday, 08 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Tuesday, 07 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Monday, 06 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Friday, 03 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Thursday, 02 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Wednesday, 01 September 2021

• A conference in celebration of the work of Bill Crawley-Boevey

Friday, 16 July 2021

• 13:15, Zoom
  Joseph Chuang (London): Rank functions on triangulated categories
  Abstract: A rank function is a nonnegative real-valued, additive, translation-invariant function on the objects of a triangulated category satisfying the triangle inequality on distinguished triangles. Rank functions on the perfect derived category of a ring are related to Sylvester rank functions on finitely presented modules, and therefore, via the work of Cohn and Schofield, to representations of the ring over skew fields. In my talk I will focus on examples. This is joint work with Andrey Lazarev.

Friday, 09 July 2021

• 13:15, Zoom
  Paolo Stellari (Milano): Uniqueness of enhancements: derived and geometric categories
  Abstract: In this talk we address several open questions and generalize the existing results about the uniqueness of enhancements for triangulated categories which arise as derived categories of abelian categories or from geometric contexts. If time permits, we will also discuss applications to the description of exact equivalences. This is joint work with A. Canonaco and A. Neeman.

Friday, 25 June 2021

• 13:15, Zoom
  Martin Kalck: A surface and a threefold with equivalent singularity categories
  Abstract: We start with an introduction to singularity categories and equivalences between them. In particular, we recall known results about singular equivalences between commutative rings, which go back to Knörrer, Yang, Kawamata and a joint work with Karmazyn. Then we explain a new singular equivalence between an affine surface and an affine threefold. This seems to be the first (non-trivial) example of a singular equivalence involving rings of even and odd Krull dimension.

Friday, 18 June 2021

• 13:15, Zoom
  Norihiro Hanihara (Nagoya): Yoneda algebras from additive generators
  Abstract: Yoneda algebras form a class of algebras which have widely been studied in ring theory and representation theory. They are defined for a ring A and an A-module M as the direct sum of Ext^i_A(M,M) over all i endowed with the Yoneda product. We discuss these Yoneda algebras in the following setting: A is a finite dimensional algebra of finite representation type, and M is the additive generator for the category of A-modules. We will give some fundamental results on such Yoneda algebras, such as coherence, Gorenstein property, periodicity, and a description of the stable category of Cohen-Macaulay modules.

Friday, 11 June 2021

• 13:15, Zoom
  Job Rock (Boston): Composition series of arbitrary cardinality
  Abstract: We discuss a generalization of the notion of a composition series in an abelian category to one of arbitrary cardinality. Then we discuss sufficient axioms that yield "Jordan—Hölder—Schreier like" theorems. Examples of these settings include pointwise finite-dimensional persistence modules and Prüfer modules. We will conclude with evidence that suggests the axioms are necessary for our "Jordan—Hölder—Schreier like" theorems. This is joint work with Eric J. Hanson.

Friday, 30 April 2021

• 13:15, Zoom
  Sondre Kvamme (Uppsala): Admissibly presented functors
  Abstract: Wanting to extend the functorial approach of Auslander to exact categories, we introduce the category of admissibly presented functors mod_{adm}(E) for an exact category E. Using this category, we extend Auslanders formula from abelian to exact categories. Furthermore, we characterize exact categories equivalent to categories of the form mod_{adm}(E), and we show that they have properties similar to module categories of Auslander algebras. For a fixed idempotent complete category C, we use this construction to show that exact structures on C are in bijection with certain resolving subcategories of mod C, and we compare this with the bijection to certain Serre subcategories of mod C due to Enomoto. This is joint work with Ruben Henrard and Adam-Christiaan Van Roosmalen.
  

Friday, 23 April 2021

• 13:15, Zoom
  Rene Marczinzik (Stuttgart): Homological algebra and combinatorics
  Abstract: We show that the incidence algebra of a finite lattice L is Auslander regular if and only if L is distributive. As an application we show that the order dimension of L coincides with the global dimension of its incidence algebra when L has at least two elements and we give a categorification of the rowmotion bijection for distributive lattices. At the end we discuss the Auslander regular property for other objects coming from combinatorics. This is joint work with Osamu Iyama.
  We also report on recent joint work with Aaron Chan, Erik Darpö and Osamu Iyama on fractionally Calabi-Yau algebras and their trivial extension algebras with relations to combinatorics and lattices.

Friday, 16 April 2021

• 14:15, Zoom
  Xiao-Wu Chen (Hefei): Skew group categories, algebras associated to Cartan matrices and folding of root lattices
  Abstract: The folding of root lattices is fundamental in Lie theory when getting from the simply-laced cases to the non-simply-laced cases. Following Gabriel and Geiss-Leclerc-Schroer, the relevant root lattices are categorified by certain module categories. We obtain a categorification of the folding projection, namely a certain functor between the module categories whose K_0-shadow is the folding projection. The main tools are skew group categories and finite EI categories of Cartan type. This is joint with Ren Wang at USTC.

Friday, 12 February 2021

• 15:15, Zoom
  Martin Gallauer (Oxford): Cohomological singularity
  Abstract: Let M be a representation of a finite group with coefficients in a ring. I want to discuss the following slogan: M is controlled by permutation modules if and only if its cohomology is non-singular. This is joint work with Paul Balmer whose talk at ICRA2020 I intend to complement with this discussion.

Saturday, 30 January 2021

• 16:15, Zoom
  Joshua Hunt: Decompositions of the stable module ∞-category
  Abstract: The Picard group of the stable module category of a finite group G has been studied by representation theorists as the group of "endotrivial modules". In this talk, I will outline an approach to studying endotrivial modules via descent, using the fact that we can decompose the stable module category as a limit of ∞-categories. This is joint work with Tobias Barthel and Jesper Grodal.

Friday, 22 January 2021

• 15:15, Zoom
  Gustavo Jasso (Bonn): Universal properties of derived categories, after Lurie
  Abstract: Let G be a Grothendieck category. The derived category D(G) of G and the homotopy category K(Inj G) of complexes of injective objects in G play important roles in representation theory and algebraic geometry. In this talk I will explain–following Lurie–the strong universal properties enjoyed by the infty-categorical refinements of D(G) and K(Inj G) as well as how these relate to the construction of realisation functors and derived equivalences.
  Knowledge of infinity-category theory will not be assumed.

Friday, 15 January 2021

• 15:15, Zoom
  Teresa Conde (Stuttgart): Quasihereditary algebras with exact Borel subalgebras
  Abstract: Exact Borel subalgebras of quasihereditary algebras emulate the role of "classic" Borel subalgebras of complex semisimple Lie algebras. Not every quasihereditary algebra A has an exact Borel subalgebra. However, a theorem by Koenig, Külshammer and Ovsienko establishes that there always exists a quasihereditary algebra Morita equivalent to A that has a (regular) exact Borel subalgebra. Despite that, an explicit characterisation of such "special" Morita representatives is not directly obtainable from Koenig, Külshammer and Ovsienko's work. In this talk, I shall present a numerical criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra and I will provide a method to compute all Morita representatives of A that have a regular exact Borel subalgebra. We shall also see that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra A only depends on the composition factors of the standard and costandard A-modules and on the dimension of the Hom-spaces between standard A-modules. I will conclude the talk with a characterisation of the basic quasihereditary algebras that admit a regular exact Borel subalgebra.

Friday, 18 December 2020

• 17:15, Zoom
  Benjamin Briggs (Salt Lake City): Stable invariance of structures on Hochschild Cohomology
  Abstract: If two self-injective finite dimensional algebras are connected by a stable equivalence of Morita type then their Hochschild Cohomology algebras are isomorphic in positive degrees.
  In characteristic p this positive part of Hochschild cohomology is actually a restricted graded Lie algebra. Since the restricted structure is nonlinear it can be difficult to handle functorially, and so Linckelmann asked whether this too passes across a stable equivalence of Morita type. I'll talk about sone work with Rubio y Degrassi where we answer this question and give a few applications.
  If there’s time I’ll talk about ongoing work with Rubio y Degrassi and Saorín which connects this with the fundamental group of a finite dimensional algebra.

Friday, 11 December 2020

• 14:30, Zoom
  Rudradip Biswas (Manchester): Generation of derived and stable categories for groups in Kropholler's hierarchy
  Abstract: We will look at the generation of a range of derived categories of modules over groups in Peter Kropholler's hierarchy. For this, we will be mostly using the language of generation in triangulated categories with localizing and colocalizing subcategories. We'll then look at a range of interesting applications and questions related to these results. And finally, we will prove a couple of generation properties of the stable module category of groups in these hierarchies that admit complete resolutions. Such stable module categories (note that the groups here need not be finite) have only recently been defined and studied by Mazza and Symonds.

Friday, 04 December 2020

• 14:30, Lecture Hall H12
  Greg Stevenson (Glasgow): Differential graded algebras with finite dimensional cohomology
  Abstract: It is not necessarily the case that a dg algebra with finite dimensional cohomology is quasi-isomorphic to one which is honestly finite dimensional. However, in the case that the cohomology is concentrated in negative cohomological degrees there is always a finite dimensional model. I'll explain how to prove this, give some consequences, and (probably) present some open questions.

Friday, 27 November 2020

• 14:15, Lecture Hall H12
  Eike Lau (Bielefeld): Balmer spectra of certain Deligne-Mumford stacks
  Abstract: Let G be a finite group and k a field. It is classically known that prime tensor ideals in the bounded derived category of finite kG-modules correspond to homogeneous prime ideals in the cohomology ring of G with coefficients in k. Assume that A is a commutative ring on which G acts. We consider the corresponding question for representations of G on finite projective A-modules, or equivalently for perfect complexes on the stack quotient of Spec A by the action of G.
  

Friday, 30 October 2020

• 14:15, Zoom
  Ulrich Thiel (Kaiserslautern): Introduction to Soergel Bimodules
  Abstract: Motivated by a recent publication, I'll give a gentle overview of the theory of Soergel bimodules, only assuming a little background and ignoring many details.
  

Friday, 17 July 2020

• 14:15, Zoom
  Mads Hustad Sandøy (Trondheim): Higher Koszul duality and connections with n-hereditary algebras
  Abstract: Introduced by Iyama and others, n-hereditary algebras are an attempt to generalize the good properties of hereditary algebras to algebras of higher global dimension, and they come in two flavours: n-representation finite and n-representation infinite. For an n-representation infinite algebra satisfying some assumptions, there exists an equivalence between the stable graded category of its trivial extension and the bounded derived category of a category associated to its (n+1)-th preprojective algebra. This equivalence is reminiscent of the BGG-correspondence, which is itself known to descend from Koszul duality.
  In this talk, based on joint work with Johanne Haugland, we describe how a higher version of the generalized Koszul duality introduced by Madsen and Green, Reiten and Solberg can be used to explain the existence of the aforementioned equivalence. Moreover, we describe results showing more general connections between n-hereditary algebras and certain kinds of Frobenius algebras satisfying higher Koszul duality.

Friday, 19 June 2020

• 14:15, Zoom
  Srikanth Iyengar (Salt Lake City): The Nakayama functor and its completion for Gorenstein algebras
  Abstract: Buchweitz proved a duality theorem for the singularity category of certain noetherian algebras that are Iwanaga-Gorenstein and have isolated singularities. This result, which appears in his unpublished manuscript on maximal Cohen-Macaulay modules, unifies and extends two duality theorems of Auslander, one for artin algebras and one for local commutative Gorenstein rings. The starting point of this project, which is in collaboration with Henning Krause, was to explore what impact the Gorenstein property has on the singularity category of a ring that does not necessarily have isolated singularities. Grothendieck’s duality theorem for commutative algebra points to a path for such an exploration. This path has lead us to a notion of Gorenstein algebras, broader than the one considered by Buchweitz, and to the Nakayama functor and its lift to homotopy categories associated to the algebra. The plan for my talk is to describe some of what we found.
  

Friday, 12 June 2020

• 14:15, Zoom
  Zhengfang Wang (Stuttgart): On Keller’s conjecture for singular Hochschild cohomology
  Abstract: Very recently, Keller shows that for a noetherian algebra A whose bounded dg derived category is smooth, the singular Hochschild cohomology (= Tate-Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category of A. He conjectures that the above isomorphism lifts to an isomorphism in the homotopy category of B-infinity algebras at the complex level. In this talk, we will show a weakened version of this conjecture for algebras with radical square zero via Krause’s stable derived category. We will also show that Keller’s conjecture is invariant under one-point (co)extensions and certain singular equivalences. This is an ongoing joint work with Xiaowu Chen and Huanhuan Li.

Friday, 05 June 2020

• 14:15, Zoom
  Nicolas Berkouk (Paris): Derived methods towards stable invariants for persistence
  Abstract: Multi-parameter persistence modules can be thought of as graded-modules over a certain polynomial ring. Nothing new under the sun one can be tempted to say. However, topological data analysis has shed a new light on these objects by introducing a notion of distance between them (the interleaving distance), which is crucial for applications. One important direction of research in this field is to seek for computable invariants of multi-parameter persistence modules which are stable with respect to this metric. Unfortunately, classical invariants from combinatorial algebraic geometry do not satisfy any form of stability in the general case. In this talk, we will illustrate how to equip the homotopy category of persistence modules with an interleaving distance, which allows to perform homological operations (such as taking minimal free resolutions) in a « stable » way. If times allow, I will also explain how to relate persistence modules and the interleaving distance with sheaves on a real vector space and the convolution distance (which was introduced by Kashiwara-Schapira). This last part is joint with François Petit.

Friday, 22 May 2020

• 14:15, Zoom
  Vincent Gelinas (Dublin): The depth, the delooping level and the finitistic dimension
  Abstract: Our knowledge of the finitistic dimension varies drastically between the settings of Artin algebras and of commutative local Noetherian rings. In the commutative case, the Auslander-Buchsbaum formula shows it equals the depth of the ring, while no similar such result holds in the noncommutative case. In this talk, we will introduce a new invariant called the “delooping level” for Noetherian semiperfect rings, which is implicit in some proofs of the Auslander-Buchsbaum formula. We show that it recovers the depth in the commutative local case, and that it provides an upper bound for the finitistic dimension of Artin algebras, with agreement under certain cohomological conditions. This is based on the preprint arxiv:2004.04828.

Friday, 15 May 2020

• 14:15, Zoom
  Sira Gratz (Glasgow): SL(k)-friezes
  Abstract: Classical frieze patterns are combinatorial structures which relate back to Gauss' Pentagramma Mirificum, and have been extensively studied by Conway and Coxeter in the 1970's.
  A classical frieze pattern is an array of numbers satisfying a local (2 x 2)-determinant rule. Conway and Coxeter gave a beautiful and natural classification of SL(2)-friezes via triangulations of polygons. This same combinatorics occurs in the study of cluster algebras, and has revived interest in the subject. From this point of view, a natural way to generalise the notion of a classical frieze pattern is to ask of such an array to satisfy a (k x k)-determinant rule instead, for k bigger than 2, leading to the notion of higher SL(k)-friezes. While the task of classifying classical friezes yields a very satisfying answer, higher SL(k)-friezes are not that well understood to date.
  In this talk, we'll discuss how one might start to fathom higher SL(k)-frieze patterns. The links between SL(2)-friezes and triangulations of polygons suggests a link to Grassmannian varieties under the Plücker embedding. We find a way to exploit this relation for higher SL(k)-friezes, and provide an easy way to generate a number of SL(k)-friezes via Grassmannian combinatorics, and suggest some ideas towards a complete classification using the theory of cluster algebras.
  This talk is based on joint work with Baur, Faber, Serhiyenko and Todorov.
  

Friday, 08 May 2020

• 14:15, Zoom
  Manuel Flores Galicia (Bielefeld): Quasi-hereditary structures on path algebras of types A, D and E
  Abstract: A quasi-hereditary algebra is an Artin algebra A together with a partial order on its set of isoclasses of simple modules satisfying certain conditions. A quasi-hereditary structure is an equivalence class of partial orders for an appropriate equivalence relation, all together giving rise to what we call the poset of quasi-hereditary structures on A. In this talk we will provide a full description of the poset of quasi-hereditary structures on a path algebra of Dynkin type A. For types D and E, we give a counting method for the number of quasi-hereditary structures. This talk is based on a joint paper with Yuta Kimura and Baptiste Rognerud (arXiv:2004.04726v2).

Friday, 31 January 2020

• 13:15, Room V2-200
  Sophiane Yahiatene (Bielefeld): Hurwitz action in extended Weyl groups with application to hereditary abelian categories
  Abstract: The main object of this talk is the family of extended Weyl groups. These are reflection groups attached to categories of coherent sheaves over a weighted projective line. After stating important properties of these groups we focus on the set of minimal reflection factorizations of a class of distinguished elements, called Coxeter transformations. We prove that in almost all cases the so-called Hurwitz action is transitive on it. Afterwards, if time permitting we point out the connection to thick subcategories generated by exceptional sequences.
  (Joint work with B. Baumeister and P. Wegener)
• 14:30, Room V2-200
  Martin Kalck (Freiburg): Relative singularity categories: dg-models and applications
  Abstract: We'll try to explain how relative singularity categories can be used to gain insight into the structure of singularity categories. A key technique is a construction describing the relative singularity category as the perfect derived category of an "explicit" dg algebra.
  We will explain this technique and show how it can be applied to unify and extend some results on singularity categories.
  This talk is based on joint work with Dong Yang.
  
• 16:00, Room V2-200
  Tashi Walde (Bonn): Homotopy coherent theorems of Dold—Kan type
  Abstract: The classical Dold—Kan correspondence is an equivalence of categories between connective chain complexes and simplicial objects in any abelian category. It is often implicit in key homological constructions such as the bar resolution and in fact forms the fundamental link between homotopical and homological algebra. In the last decades many variants of the Dold—Kan correspondence have been established and various axiomatic frameworks have been proposed to tie these equivalences together. Both in algebra and topology, situations arise where it is useful to work homotopy coherently (aka “derived"), i.e. study diagrams which don't just commute on the nose, but only up to specified (possibly higher) homotopies; the main examples being derived categories, stable module categories or stable homotopy theory.
  In this talk we explain the key feature of the aforementioned "theorems of Dold—Kan type” and how they can be generalized to the homotopy coherent context by employing the theory of infinity-categories.

Monday, 27 January 2020

• 16:15, Room T2-227
  Dirk Kussin (Berlin): Cotilting sheaves and indecomposable pure-injective sheaves over noncommutative curves
  Abstract: A quasicoherent sheaf is - by definition - of slope infinity, if it does not map to vector bundles. We classify all cotilting sheaves of slope infinity and all indecomposable pure-injective sheaves of slope infinity over any weighted noncommutative regular projective curve. (This includes all smooth projective curves, all weighted projective curves, etc.) This leads in the special cases where the curve is domestic, tubular or elliptic to a complete description of all non-coherent cotilting sheaves, and to a classification of the indecomposable pure-injective sheaves. In the tubular/elliptic cases we also address some still open problems for the irrational slopes. This is joint work with Rosanna Laking.

Friday, 17 January 2020

• 14:15, Room V2-200
  Daniel Labardini-Fragoso (Mexico City): Algebraic and combinatorial decompositions of Fuchsian groups
  Abstract: The discrete subgroups of the real projective special linear group of degree two are often called 'Fuchsian groups'. For a Fuchsian group G whose action on the hyperbolic plane H is free, the orbit space H/G has a canonical structure of Riemann surface with a hyperbolic metric, whereas if the action of G is not free, then H/G has a structure of 'orbifold'. In the former case, there is a direct and very clear relation between G and the fundamental group of H/G: a theorem of the theory of covering spaces states that they are isomorphic. When the action of G is not free, the relation between G and the fundamental group of H/G is more subtle. A 1968 theorem of Armstrong states that the fundamental goup is the quotient of G modulo the subgroup E generated by elliptic elements. For G finitely generated, non-elementary and with at least one parabolic element, I will present full algebraic and combinatorial decompositions of G in terms of the fundamental group of H/G and a specific finitely generated subgroup of E, thus improving Armstrong's theorem.
  This talk is based on an ongoing joint project with Sibylle Schroll and Yadira Valdivieso-Díaz that aims at describing the bounded derived categories of skew-gentle algebras in terms of curves on surfaces with orbifold points of order 2.
• 15:30, Room V2-200
  Igor Burban (Paderborn): Homological mirror symmetry for compact surfaces with boundary, tame non-commutative nodal curves and spherical objects on cycles of projective lines
  Abstract: In my talk, I am going to explain, how tame non-commutative nodal curves appear as "holomorphic mirrors" of certain graded compact oriented surfaces with marked boundary. This part of the talk is based on a work of Yanki Lekili and Alexander Polishchuk, as well as on my joint works with Yuriy Drozd.
  As a nice application of the homological mirror symmetry, I am going to explain the classification of spherical objects on a cycle of projective lines. This part of my talk is based on a part of the PhD thesis of Sebastian Opper.

Friday, 10 January 2020

• 13:15, Room V2-200
  Xiao-Wu Chen (Hefei): Leavitt path algebras, B-infinity algebras and Keller's conjecture
  Abstract: Recently, Keller proves that the Tate-Hochschild cohomology algebra is isomorphic to the Hochschild cohomology algebra of the dg singularity category. He conjectures that the isomorphism lifts a B-infty-isomorphism on the cochain level. We verify his conjecture for an algebra with radical square zero, using the corresponding Leavitt path algebra. One ingredient of the proof is to enhance Krause's description of the singularity category to the dg level. This is joint with Zhengfang Wang and Huanhuan Li.
• Bielefeld-Paderborn-Seminar
  14:30, Room V2-200
  Henning Krause (Bielefeld): Local versus global for representations of algebras
  Abstract: We consider some classes of finite dimensional algebras and discuss the classification of thick subcategories for their module categories. Typical examples are path algebras of quivers or group algebras of finite groups. This leads naturally to the study of derived categories. When a cohomology ring is acting, we may pass from global to local and obtain in some interesting cases a stratification of the module category.
• Bielefeld-Paderborn-Seminar
  16:30, Room V2-200
  Kai-Uwe Schmidt (Paderborn): Highly nonlinear functions
  Abstract: The nonlinearity of a Boolean function in n variables is its Hamming distance to the set of all affine Boolean functions in n variables. Boolean functions with large nonlinearity are difficult to approximate by affine Boolean functions, which is of significant interest in cryptography. The largest possible nonlinearity of a Boolean function in n variables also equals the covering radius of the [2^n,n+1] Reed-Muller code, whose determination is subject to a famous conjecture from the 1980s.
  In this talk, I will survey the history of this conjecture and then explain how the conjecture can be proved using a mixture of number-theoretic and probabilistic arguments. I will also discuss generalisations of this conjecture.

Friday, 13 December 2019

• 13:15, Room V2-200
  Maitreyee Kulkarni (Bonn): Infinite friezes and triangulations of an annulus
  Abstract: In this talk I will introduce a combinatorial object called a frieze and describe its relations to triangulations and to representations of certain quivers. In particular, we will see that each periodic infinite frieze determines a triangulation of an annulus in a unique way. We will also study associated module categories and determine an invariant of friezes in terms of modules. This is joint work with Karin Baur, Ilke Canakci, Karin Jacobsen, and Gordana Todorov.
  
• 14:30, Room V2-200
  Baptiste Rognerud (Paris): Equivalences between blocks of cohomological Mackey algebras
  Abstract: Mackey functors were introduced as a convenient tool for handling the induction theory of several objects having a similar behavior (group representations; representation rings, group cohomology, etc;). Later, it was proved by Thévenaz and Webb that the category of Mackey functors is equivalent to the category of modules over a finite dimensional algebra called the Mackey algebra. The proof is far from being difficult, but this result is of crucial importance : one can study Mackey functors using the ring and module theory. It turns out that the Mackey algebra is, in many aspects, similar to the group algebra.
  In this talk, I will explain how the problems of constructing (derived) equivalences between categories representations of finite groups and between the corresponding categories of cohomological Mackey functors are related. We will see that the easy situation of Morita equivalences between blocks of group algebras may be much more interesting in the world of Mackey functors. This is part of a joint work with Markus Linckelmann.

Friday, 29 November 2019

• 14:15, Room V2-200
  Pedro Fernández (Bogotá): Matrix problems associated to some Brauer configuration algebras
  Abstract: Bijections between solutions of the Kronecker problem and the four subspace problem with indecomposable projective modules over some Brauer configuration algebras are obtained by interpreting elements of some integer sequences as polygons of suitable Brauer configurations. This kind of configurations are also used to categorify (in the sense of Ringel and Fahr) some integer sequences.
  This is joint work with Agustín Moreno Cañadas.
• 15:30, Room V2-200
  Helmut Lenzing (Paderborn): Algebraic theory of fuchsian singularities
  Abstract: Fuchsian singularities are graded isolated singularities of Krull dimension two. Classically, they arise as rings of G-invariant differential forms (automorphic forms) with respect to a fuchsian group, a discrete cocompact subgroup G of the automophism group of the hyperbolic plane of complex numbers with positive imaginary part.
  My talk has the following aims: (1) Generalize the concept of fuchsian singularities to algebraically closed fields of arbitrary characteristic. (2) Relate them to mathematical objects of a different nature. (3) Provide a purely ring theoretic characterization of fuchsian singularities.
  We further determine their singularity categories together with relevant Grothendieck group related data.
  

Friday, 22 November 2019

• 13:15, Room V2-200
  Jenny August (Bonn): The Stability Manifold of a Contraction Algebra
  Abstract: For a finite dimensional algebra, Bridgeland stability conditions can be viewed as a continuous generalisation of tilting theory, providing a geometric way to study the derived category. Describing this stability manifold is often very challenging but in this talk, I’ll look at a special class of symmetric algebras whose tilting theory is determined by a related hyperplane arrangement. This simple picture will then allow us to describe the stability manifold of such an algebra.
• 14:30, Room V2-200
  Magnus Hellstrøm-Finnsen (Trondheim): The spectrum for an additive and an exact monoidal category
  Abstract: We will define and investigate some basic properties of the spectrum for an additive and a Quillen exact monoidal category. Further we will define a notion of support data on these categories and classify radical ideals with supported by primes.
  
• 16:00, Room V2-200
  Gustavo Jasso (Bonn): The symplectic geometry of higher Auslander algebras
  Abstract: It follows from work of Bocklandt, Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk that (appropriate versions of) the Fukaya categories associated to marked Riemann surfaces are equivalent to (appropriate versions of) the derived categories of graded locally gentle algebras.
  In this talk I will explain the first steps in a higher-dimensional generalisation of the above. Natural higher-dimensional symplectic manifolds associated to Riemann surfaces are their symmetric products. In the simplest case of a marked disk, I will detail a description of the (partially wrapped) Fukaya categories of its symmetric products in terms of the derived categories of Iyama's higher-dimensional Auslander algebras of type A. Intrincate combinatorics (observed first by Auroux and Lipshitz-Ozsváth-Thurston) related to the Bruhat order of the symmetric group arise already in these simplest higher-dimensional examples.
  This is a report on joint work with Tobias Dyckerhoff and Yanki Lekili.
  

Friday, 15 November 2019

• Bielefeld-Paderborn Seminar
  14:15, Room V2-200
  William Crawley-Boevey (Bielefeld): Clannish algebras revisited
  Abstract: We are concerned with classifying the finitely generated indecomposable modules for a finite-dimensional associative algebra, or more generally a ring, or some related situation, such as objects in a derived category. There are a number of situations where classifications have been obtained in terms of so-called strings and bands. This includes string algebras, clannish algebras (introduced by the speaker in 1989), Dedekind-like rings and nodal algebras. I shall review some of this work, with examples from geometry, topology and arithmetic. In addition, I aim to describe some improvements to my earlier work on clannish algebras.
• Bielefeld-Paderborn Seminar
  15:30, Room V2-200
  Fabian Januszewski (Paderborn): A cohomological approach to characters of Lie groups

Friday, 25 October 2019

• 14:15, Room V2-200
  Fei Xie (Bielefeld): The derived category of a singular quintic del Pezzo surface
  Abstract: I will give a semiorthogonal decomposition for the bounded derived category of coherent sheaves on a quintic del Pezzo surface with mild singularity (rational Gorenstein) over algebraically closed fields. The decomposition has three components. Two components are equivalent to derived categories of the base field. The remaining component is equivalent to the derived category of products of truncated polynomials with total length 5. The decomposition is obtained by studying the semiorthogonal decomposition of the minimal resolution of the surface. I will also briefly mention how to obtain a similar decomposition using Homological Projective Duality and how to obtain a decomposition over non-algebraically closed fields.
• 15:30, Room V2-200
  Paul Wedrich (Bonn): Quivers for SL(2) tilting modules
  Abstract: I will explain how diagrammatic algebra can be used to give an explicit generators-and-relations presentation of all morphisms between indecomposable tilting modules for SL(2) over an algebraically closed field of positive characteristic. The result takes the form of a path algebra of an infinite, fractal-like quiver with relations, which can be considered as the (semi-infinite) Ringel dual of SL(2). Joint work with Daniel Tubbenhauer.

Friday, 18 October 2019

• 14:15, Room V2-200
  Alexander Slávik (Prague): On flat generators and Matlis duality for quasicoherent sheaves
  Abstract: We show that for a quasicompact quasiseparated scheme X, the following assertions are equivalent: (1) the category QCoh(X) of all quasicoherent sheaves on X has a flat generator; (2) for every injective object E of QCoh(X), the internal hom functor into E is exact; (3) the scheme X is semiseparated. Joint work with Jan Šťovíček.

Thursday, 26 September 2019

• Workshop "Representation Theory in Bielefeld – Past and Future"

Wednesday, 25 September 2019

• Workshop "Representation Theory in Bielefeld – Past and Future"

Tuesday, 24 September 2019

• Workshop "Representation Theory in Bielefeld – Past and Future"

Monday, 16 September 2019

• 14:15, Room V5-227 (60 minutes)
  Marc Stephan (Bonn/Augsburg): Interactions between elementary abelian p-group actions in topology and algebra
  Abstract: I will provide a selective overview about rank conjectures for actions of elementary abelian p-groups. They estimate the size of the total dimension in homology over a field of characteristic p for free actions on finite CW complexes or finite chain complexes. Recently, Iyengar and Walker found algebraic examples with smaller homology than predicted, while joint work with Henrik Rüping shows that these counterexamples can not be realized topologically.
  To establish bounds for the total dimension in homology, it is still interesting to consider the algebraic version and connect it to problems in commutative algebra. I will explain such a connection from joint work with Jeremiah Heller, and how it is related to constructions of vector bundles on projective space from modules of constant Jordan type due to Benson and Pevtsova.

Monday, 09 September 2019

• BiBo Seminar in Bielefeld
  11:15, Room U2-200 (60 minutes)
  Biao Ma (Bielefeld): Combinatorics of faithfully balanced modules
  Abstract: In this talk, I will give a combinatorial characterization of faithfully balanced modules for the path algebra of the quiver An with linear orientation. By using this characterization one can deduce that the number of basic faithfully balanced modules is the nth 2-factorial number. Among them are n! modules with exactly n indecomposable summands which form a lattice (with respect to some appropriate partial order) – this extends the lattice of tilting modules. This is joint work with William Crawley-Boevey, Baptiste Rognerud and Julia Sauter.
• Bielefeld-Bochum-Seminar
  13:30, Room U2-200 (60 minutes)
  Alexander Pütz (Bochum): Degenerate Affine Flag Varieties and Quiver Grassmannians
  Abstract: We study degenerate flag varieties where certain projections replace the identity maps in the inclusion relations for the chains of the spaces in the geometric interpretation of the flag variety. Quiver Grassmannians are projective varieties parametrising subrepresentations of a quiver representation.
  We show that certain quiver Grassmannians for the equioriented cycle provide finite dimensional approximations of the degenerate affine flag variety of type GL_n. These quiver Grassmannians admit a finite cellular decomposition parametrised by affine Dellac configurations. Their irreducible components are normal, Cohen-Macaulay, have rational singularities and are parametrised by grand Motzkin paths. The Poincaré polynomials of the approximations admit a description based on the affine Dellac configurations. This research links the theory of quiver Grassmannians with the representation theory of affine Kac-Moody groups.
• Bielefeld-Bochum-Seminar
  15:00, Room U2-200 (30 minutes)
  Julia Sauter (Bielefeld): Relative geometry of representations - II
  Abstract: We have a closer look at F-hereditary structures. Every algebra of representation dimension at most three admits an F-hereditary structure. All relative representation spaces are smooth if and only if the relative structure is F-hereditary. Furthermore, for relative quiver Grassmannians we can show that they are smooth if we have an F-hereditary structure and an F-rigid module.

Friday, 05 July 2019

• 14:15, Room T2-208
  Estanislao Herscovich (Grenoble): The cohomology of the Fomin-Kirillov algebra on 3 generators
  Abstract: The aim of the talk is to present an elementary computation of the algebra structure of the Yoneda algebra of the Fomin-Kirillov algebra on 3 generators, based on a bootstrap technique built over the (non-acyclic) Koszul complex.
• 15:30, Room T2-208
  Grzegorz Bobiński (Torun): A characterization of representation infinite quiver settings
  Abstract: We characterize pairs (Q,d) consisting of a quiver Q and a dimension vector d, such that over a given algebraically closed field k there are infinitely many representations of Q of dimension vector d. We also present an application of this result to the study of algebras with finitely many orbits with respect to the action of (the double product) of the group of units.
  

Friday, 07 June 2019

• 14:15, Room T2-208
  Philipp Lampe (Canterbury): The growth of real seeds and a determinant from group representation theory
  Abstract: This talk provides a taster of an ingredient that we added to a proof in a joint work with Anna Felikson. In particular, we will look at a determinant assembled from the characters of a finite group. First, we give an overview of its long and colourful history going back to Catalan, Dedekind, Burnside and Frobenius. Second, we explain how the determinant helped to estimate growth rates of real seeds in cluster theory.

Friday, 24 May 2019

• 14:15, Room T2-208
  Janina Carmen Letz (Salt Lake City): Local to global principles for generation time over Noether algebras
  Abstract: In the derived category of modules over a Noether algebra a complex G is said to generate a complex X if the latter can be obtained from the former by taking finitely many summands and cones. The number of cones needed in this process is the generation time of X. I will present some local to global type results for computing this invariant, and also discuss some applications.
• 15:30, Room T2-208
  Zhengfang Wang (Bonn): Tate-Hochschild cohomology and B-infinity algebra
  Abstract: Tate-Hochschild cohomology was implicitly defined in Buchweitz’ unpublished manuscript in 1986, using his stable derived category. Analogous to Hochschild cohomology, it is interesting to ask whether there is a Gerstenhaber algebra structure on Tate-Hochschild cohomology.
  In this talk, we will give an affirmative answer to the above question. For this, we first construct a natural complex to compute Tate-Hochschild cohomology. Then we show that there is a so-called B-infinity algebra structure on this complex by giving an explicit action of the little 2-discs operad on it. In particular, passing to cohomology, we get a Gerstenhaber algebra structure. If time permits, we will also talk about Keller’s very recent result and conjecture.

Friday, 17 May 2019

• Bielefeld-Münster Representation Theory Seminar
  13:15, Room T2-208
  Biao Ma (Bielefeld): Faithfully balanced modules and applications in relative homological algebra
  Abstract: For a finite-dimensional algebra we revisit faithfully balanced modules and introduce the relative version of them. As applications, we establish the relative version of Brenner-Butler's tilting theorem and (higher) Auslander correspondence. Examples will be given to explain the main results. This is joint work with Julia Sauter.
• Bielefeld-Münster Representation Theory Seminar
  14:30, Room T2-208
  Haydee Lindo (Williamstown, MA): Endomorphism invariant modules and ring classifications
  Abstract: I will speak on modules which are invariant under endomorphisms of their envelopes. This will include connections to the general theory of trace modules with some preliminary applications to ring classifications and conjectures involving modules with no self-extensions.
• Bielefeld-Münster Representation Theory Seminar
  16:00, Room T2-208
  Lutz Hille (Münster): Tilting Modules for the Auslander Algebra with a View to Derived Categories
  Abstract: We consider the Auslander algebra of the truncated polynomial ring and classify exceptional modules and spherical modules. Using a recent result of Geuenich, we can describe all tilting modules as universal extensions of full exceptional sequences. Then we use spherical twists to construct also tilting complexes in the derived category, which have a very explicit description.
  It is still open in general whether this is already the full classification, so we discuss the known results and the open problems.
  This is joint work with David Ploog.

Saturday, 04 May 2019

• Workshop "Computational Applications of Quiver Representations: TDA and QPA"

Friday, 03 May 2019

• Workshop "Computational Applications of Quiver Representations: TDA and QPA"

Thursday, 02 May 2019

• Workshop "Computational Applications of Quiver Representations: TDA and QPA"

Friday, 12 April 2019

• 13:30, Room T2-208
  Markus Linckelmann (London): On Picard groups of finite group algebras
  Abstract: The Picard group of self Morita equivalences of a finite-dimensional algebra over an algebraically closed field k is an algebraic group. By contrast, the Picard group of a finite group algebra over a p-adic ring with finite residue field is a finite group. The structure of the automorphism group of a finite group algebra over a p-local domain with an algebraically closed residue field is largely unknown; it seems to be unknown in general whether this group is finite. A recent result by F. Eisele shows that this group is also an algebraic group over the residue field. In joint work with R. Boltje and R. Kessar, we identify a `large' subgroup of the Picard group of a block algebra in terms of the fusion systems of the blocks and the Dade groups of its defect groups. This is partly motivated by the - to date open - question whether Morita equivalent block algebras have isomorphic defect groups and fusion systems. Another motivation comes from recent work of Eaton, Eisele, and Livesey, where the above results on Picard groups play a role in the proof of special cases of Donovan's finiteness conjecture.
• 14:45, Room T2-208
  Dave Benson (Aberdeen): Some exotic symmetric tensor categories in characteristic two
  Abstract: This talk is about joint work with Pavel Etingof. A theorem of Deligne says that in characteristic zero, any symmetric tensor category "of moderate growth" admits a tensor functor to vector spaces or to super (i.e., Z/2-graded) vector spaces. In prime characteristic, this is not true, but one may ask whether there is a good list of "incompressible" symmetric tensor categories to which they they do all map. We construct an infinite ascending chain of finite symmetric tensor categories in characteristic two, all of which are incompressible. The constructions are based on the theory of tilting modules over the finite groups SL(2,2^n). Similar examples should exist in other prime characteristics, but the details have not yet been worked out.
• 16:00, Room T2-208
  Jon Carlson (Athens, Georgia): Lots of categories for the Green correspondence
  Abstract: This is joint work with Lizhong Wang and Jiping Zhang.
  The object is to establish a Green correspondence for categories of complexes of modules as well as their homotopy categories and derived categories. There is a categorical expression of the Green correspondence that is similar to a construction of Benson and Wheeler. At a key point in the constuction, we must assume that one of the categories has idempotent completion. This condition holds provided the category has countable direct summands. But under that assumption there are many categories that satisfy the hypothesis.
  

Wednesday, 06 February 2019

• 14:15, Room U2-205
  Teresa Conde (Stuttgart): Gabriel-Roiter measure and finiteness of the representation dimension
  Abstract: The induction scheme used in Roiter's proof of the first Brauer-Thrall conjecture prompted Gabriel to introduce an invariant, known as the Gabriel–Roiter measure. The usefulness of the Gabriel-Roiter measure is not limited to the first Brauer-Thrall conjecture: Ringel has used it to give new proofs of results established by Auslander in the 70's. In this talk, we use the Gabriel-Roiter measure to provide a new proof of the finiteness of the representation dimension for Artin algebras, a result originally proved by Iyama in 2002.
  

Friday, 01 February 2019

• 14:15, Room U2-205
  Severin Barmeier (Bonn): Diagrams of algebras, categories of coherent sheaves and deformations
  Abstract: Given a complex algebraic variety X, the restriction of its structure sheaf to a finite cover of affine open sets can be viewed as a diagram of (commutative) algebras. Deformations of a diagram obtained in this way correspond precisely to deformations of the category of
  (quasi)coherent sheaves as an Abelian category (after W. Lowen and M. Van den Bergh).
  We describe the higher deformation theory explicitly via L-infinity algebras for X covered by two affine opens and explain the connection to
  "classical" deformations of the complex structure and deformation quantizations by means of examples. This is joint work with Y. Frégier.
• 15:30, Room U2-205
  Wassilij Gnedin (Bochum): A homological characterization of ribbon graph orders
  Abstract: Recently, finite-dimensional algebras which can be related to surfaces have attracted a lot of interest.
  My talk is concerned with certain 'non-commutative curve singularities' arising from ribbon graphs on closed surfaces.
  These so-called ribbon graph orders can be viewed as infinite-dimensional versions of Brauer graph algebras as well as gentle algebras.
  Although defined by particular combinatorial conditions, it turns out that ribbon graph orders possess a unique blend of purely homological features (such as a Calabi-Yau property of the perfect derived category and semisimplicity of the singularity category).
  Using their homological characterization I will show that ribbon graph orders as well as certain finite-dimensional special biserial algebras are preserved under derived equivalences.

Friday, 25 January 2019

• 14:15, Room U2-205
  Sebastian Opper (Paderborn): On auto-equivalences and derived invariants of gentle algebras
  Abstract: This talk will be about derived equivalences of gentle algebras and the group of auto-equivalences of their derived categories. In joint work with Plamondon and Schroll, we attached a surface to every gentle algebra and showed that its geometry encodes of the triangulated structure of the derived category. I will explain how a derived equivalence of gentle algebras gives rise to a homeomorphism between their associated surfaces and how this leads to a complete derived invariant of gentle algebras which generalizes the combinatorial invariant of Avella-Alaminos and Geiss. Finally, I will talk about applications to groups of auto-equivalences of gentle algebras and their connection to mapping class groups.
• 15:30, Room U2-205
  Frederik Marks (Stuttgart): Flat ring epimorphisms and localisations of commutative noetherian rings
  Abstract: We study different types of localisations of a commutative noetherian ring. In particular, we are interested in the following questions: When is a flat ring epimorphism a universal localisation in the sense of Schofield? And when is such a universal localisation a classical ring of fractions? We approach these questions using the theory of support and local cohomology, and by analysing the specialisation closed subset of the spectrum associated with a flat ring epimorphism. As for the first question, we show that all flat ring epimorphisms are universal localisations when the underlying ring is either locally factorial or of Krull dimension one. If time permits, we will also comment on the situation for more general rings, which turns out to be significantly more complicated and diverse. Finally, we show that an answer to the question of when universal localisations are classical depends on the structure of the Picard group of the underlying ring. This talk is based on joint work with Lidia Angeleri Hügel, Jan Stovicek, Ryo Takahashi and Jorge Vitória.

Friday, 18 January 2019

• 14:15, Room U2-205
  Catharina Stroppel (Bonn): Semiinfinite highest weight categories
  Abstract: We briefly recall the classical highest weight categories theory (following Cline-Parshall-Scott, Donkin and Ringel) for finite dimensional algebras in a language which allows generalizations to stratified algebras and infinite situations. In particular we formulate aspects of tilting theory and Ringel duality in a semi-infinite setting. If time allows we will mention some explicit examples for this construction related to diagram algebras and categorifications.
• BGTS Colloquium
  16:15, Lecture Hall H5
  Sabine Jansen (München): Condensation, big jump and heavy tails: from phase transitions to probability
  Abstract: Ice melts, water evaporates - these are everyday experiences of phase transitions. The explanation of this macroscopic phenomenon from microscopic laws belongs to the realm of statistical physics, which treats matter as a composite system made up of many individual "agents" with random behavior. From a mathematician's point of view, a fully rigorous understanding still eludes us. The search for it leads to questions in probability that open up surprising connections: toy models for surface tension of liquid droplets build on heavy-tailed variables used in insurance mathematics; a big jump made by a random walker is a condensation phenomenon in disguise. The talk explains some of these connections and presents open problems and partial answers.

Friday, 11 January 2019

• 14:15, Room U2-205
  William Crawley-Boevey (Bielefeld): Decomposition of persistence modules
  Abstract: I shall discuss the decomposition of pointwise finite-dimensional persistence modules. A persistence module indexed by the real plane is said to be middle exact if for each rectangle in the plane, the associated 3-term exact sequence of vector spaces is exact in the middle. I shall outline a new proof of a theorem of Cochoy and Oudot classifying such middle exact modules. They arise in the study of interlevel set persistence homology, answering a question of Botnan and Lesnick. This is joint work with Magnus Botnan.
• 15:45, Room U2-205
  Jörg Schürmann (Münster): Degenerate affine Hecke algebras and Chern classes of Schubert cells
  Abstract: We explain in the context of complete flag varieties X=G/B the relation between Chern classes of Schubert cells and convolution actions of degenerate affine Hecke-algebras as in the work of Ginzburg. This is based on the Lagrangian approach via characteristic cycles. As an application we show that the two cohomological Weyl group actions constructed by Ginzburg and Aluffi-Mihalcea coincide. These Weyl group actions permute the (equivariant) Chern classes of the corresponding Schubert cells. This is joint work with P. Aluffi, L. Mihalcea and C. Su.

Friday, 23 November 2018

• BiBo Seminar in Bochum
  14:00, Room IA 1-71 (60 minutes)
  Arif Dönmez (Bochum): Moduli of representations of one-point extensions
  Abstract: We study the moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations and derive results on their geometric properties with homological methods.
• BiBo Seminar in Bochum
  15:30, Room IA 1-71 (30 minutes)
  Julia Sauter (Bielefeld): An invitation to relative geometry of representations
  Abstract: Following Auslander and Solberg, relative homological algebra replaces Ext^1 by a subfunctor Ext^1_F. In this set-up it is natural to replace the representation space of quiver representations by locally closed subsets where (certain) Hom-dimensions are fixed (their closures are usually referred to as rank varieties). I would like to use relative homological algebra to study these spaces and explain as a first step the relative Voigt's lemma. I would end with many conjectures and would hope to find some interested people who would like to work on this with me.
• BiBo Seminar in Bochum
  16:45, Room IA 1-71 (60 minutes)
  Christof Geiss (Mexico City): Real Schur roots and rigid representations
  Abstract: This is a report on joint work with B. Leclerc and J. Schröer.
  Let F be a field. In previous work we constructed, associated to a symmetrizable generalized Cartan matrix C with symmetrizer D and and orientation Ω, an 1-Iwanga-Gorenstein F-algebra H:=H(C,D,Ω), defined in terms of a quiver with relations.
  We show that the rigid indecomposable locally free H-modules are parametrized, via their rank vector, by the real Schur roots associated to (C,Ω). Moreover, if M is such a module then it is free as if module over its endomorphism ring, and this ring is of the form F[x]/(x^n) for some n. This allows us to classify the left finite bricks of H in terms of the real Schur roots associated to (C^t,Ω). The main tool to prove our results is a F[[x]]-order, which permits us to relate locally free H-modules with modules over the canonical F((x))-species associated to the combinatorial data (C,D,Ω).

Friday, 16 November 2018

• 13:15, Room U2-205
  Olaf Schnürer (Paderborn): Smoothness of derived categories of algebras
  Abstract: We report on joint work with Alexey Elagin and Valery Lunts where we prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.
• 14:30, Room U2-205
  Lidia Angeleri Hügel (Verona): Silting complexes over hereditary rings
  Abstract: I will report on joint work with Michal Hrbek. Given a hereditary ring, we use the lattice of homological ring epimorphisms to construct compactly generated t-structures in its derived category. This leads to a classification of all (not necessarily compact) silting complexes over the Kronecker algebra.

Friday, 09 November 2018

• 14:15, Room U2-205
  Kevin Coulembier (Sydney): Tensor categories in positive characteristic
  Abstract: Tensor categories are abelian k-linear monoidal categories satisfying some natural additional properties. Archetypical examples are the representation categories over affine (super)group schemes. P. Deligne gave very succinct intrinsic criteria for a tensor category to be equivalent to such a representation category, over fields k of characteristic zero. These descriptions are known to fail badly in prime characteristics. In this talk, I will present analogues in prime characteristic of these criteria, admittedly less succinct, but still intrinsic. Time permitting, I will comment on the link with a recent conjecture of V. Ostrik extending Deligne’s work in a different direction.

Friday, 19 October 2018

• 14:15, Room U2-205
  Sarah Scherotzke (Münster): The Chern character and categorification
  Abstract: The Chern character is a central construction which appears in topology, representation theory and algebraic geometry. In algebraic topology it is for instance used to probe algebraic K-theory which is notoriously hard to compute, in representation theory it takes the form of classical character theory. Recently, Toen and Vezzosi suggested a construction, using derived algebraic geometry, which allows to unify the various Chern characters. We will categorify this Chern character. In the categorified picture algebraic K-theory is replaced by the category of non-commutative motives.
• 15:30, Room U2-205
  Jörg Feldvoss (Mobile, Alabama): Cohomological Vanishing Theorems for Leibniz Algebras
  Abstract: Leibniz cohomology was introduced by Bloh and Loday as a non-commutative analogue of Chevalley-Eilenberg cohomology of Lie algebras. It turned out that Leibniz cohomology works more generally for Leibniz algebras which are a non-(anti)commutative version of Lie algebras. Many results for Lie algebras have been proven to hold in this more general context.
  In the talk I will start from scratch and define Leibniz algebras, Leibniz (bi)modules, and their cohomology. Then I will explain the Leibniz analogues of vanishing theorems for the Chevalley-Eilenberg cohomology of semisimple and solvable Lie algebras due to Whitehead, Dixmier, and Barnes. In particular, we obtain the second Whitehead lemma for Leibniz algebras and the rigidity of semisimple Leibniz algebras in characteristic zero. The latter results were conjectured to hold for quite some time. Our main tools are the cohomological analogues of two spectral sequencesof Pirashvili for Leibniz homology and a spectral sequence due to Beaudouin.
  All this is joint work with Friedrich Wagemann.

Friday, 24 August 2018

• 14:15, V2-200
  Hiroyuki Minamoto (Sakai): Finite dimensional graded Iwanaga-Gorenstein algebras and Happel's functor
  Abstract: Let A be a finite dimensional algebra and T(A)= A + D(A) be the trivial extension by the dual bimodule D(A). Happel constructed a functor H from the derived category of A to the stable category of graded T(A)-modules and proved that it is fully faithful and gives an equivalence precisely when A is of finite global dimension. Happel's functor is generalized to a finite dimensional graded Iwanaga-Gorenstein algebra S and gives a functor from the derived category of the Beilinson algebra, a finite dimensional algebra constructed from S, to the stable category of graded Cohen-Macaulay modules. In this talk, first we give two characterizations of S such that Happel's functor is fully faithful or equivalence. Next we show that Happel's functor admits a fully faithful left adjoint which has left adjoint provided that the degree 0 subalgebra of S is of finite global dimension. If time permits, we give several applications. This is a joint work with Kota Yamaura.

Friday, 20 July 2018

• 14:15, Room T2-205
  Baptiste Rognerud (Bielefeld): The derived category of the Tamari lattice is fractionally Calabi-Yau
  Abstract: In this talk, I will introduce an interesting family of indecomposable objects in the bounded derived category of the Tamari lattice. Then, I will give a combinatorial description of the action of the Serre functors on these objects and explain how we can deduce that the bounded derived category is fractionally Calabi-Yau.
• 15:30, Room T2-205
  Fan Xu (Beijing): Ringel-Hall algebras and categorification
  Abstract: The aim of this talk is to generalize Lusztig's construction of quantum groups to Ringel-Hall algebras. We construct the geometric analog of Green's theorem on the comultiplication of a Ringel-Hall algebra. It is an extension version of the comultiplication of a quantum group defined by Lusztig. As an application, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework. This is based on joint work with Jie Xiao and Minghui Zhao.

Friday, 13 July 2018

• 15:00, Room T2-205
  Louis Rowen (Ramat Gan): Algebraic systems and exterior semi-algebra
  Abstract: In this talk, we describe negation maps and `systems', and their application to linear algebra in a rather general framework that includes tropical algebra, hyperfields and fuzzy rings.
  The usual definition of Grassmann (exterior) algebras generalizes directly to semi-algebras, and has a built-in negation map for elements of degree > 1, so the theory of systems can be applied directly to Gatto's theory, unifying results of linear algebra from different perspectives including the classical perspective.
  This will include joint work with Akian, Gaubert, Gatto, Jun, Knebusch, and Mincheva, and does not require prerequisites.

Friday, 06 July 2018

• 14:15, Room T2-205
  Ulrich Thiel (Sydney): Finite-dimensional graded algebras with triangular decomposition
  Abstract: I will discuss a new approach to the representation theory of self-injective finite-dimensional graded algebras with triangular decomposition (such as restricted enveloping algebras, Lusztig’s small quantum groups, hyperalgebras, finite quantum groups, restricted rational Cherednik algebras, etc). We show that the graded module category of such an algebra is a highest weight category and has a tilting theory in the sense of Ringel. We can then show that the degree zero part of the algebra (the "core") is cellular and can construct a canonical highest weight cover à la Rouquier of it. The core captures essential information of the representation theory of the original algebra, hence we can approach the latter with these additional structures. This is joint work with Gwyn Bellamy (Glasgow).
• 15:30, Room T2-205
  Yann Palu (Amiens): Non-kissing complex and tau-tilting over gentle algebras
  Abstract: This is a report on a joint paper with Vincent Pilaud and Pierre-Guy Plamondon. The non-kissing complex is a simplicial complex introduced by T. McConville who studied some of its lattice theoretic aspects. After explaining the properties of the non-kissing complex that seem the most relevant to representation theory, I will relate it to tau-tilting theory, as defined by Adachi-Iyama-Reiten. This allows to generalise non-kissing to a more general set up by making use of gentle algebras.

Wednesday, 27 June 2018

• 14:15, Room U2-205 (90 minutes)
  Giovanni Cerulli Irelli (Rome): Cell decompositions and algebraicity of cohomology for quiver Grassmannians
  Abstract: I will report on a joint project with F. Esposito (Padova), H. Franzen (Bochum) and M. Reineke (Bochum), arXiv: 1804.07736. We show that the cohomology ring of a quiver Grassmannian associated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this, we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated to an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular decomposition for quiver Grassmannians associated with representations with a rigid regular part. Finally, we study the geometry behind the cluster multiplication formula of Caldero and Keller, providing a new proof of a slightly more general result.

Friday, 15 June 2018

• 14:15, Room U2-113
  Jan Geuenich (Bielefeld): Tilting modules for the Auslander algebra of the truncated polynomial ring
  Abstract: I present the classification of the tilting modules for the Auslander algebra of the truncated polynomial ring. More precisely, I construct an isomorphism between the tilting poset and a finite interval in the braid group. This extends an isomorphism described by Iyama and Zhang between the classical tilting poset and the symmetric group with the opposite left weak order.
• 15:30, Room U2-113
  Estanislao Herscovich (Grenoble): The curved A_infinity-coalgebra of the Koszul codual of a filtered dg algebra
  Abstract: In this talk I will present a result allowing to compute the coaugmented curved A_infinity-coalgebra structure of the Koszul codual of a filtered dg algebra over a field k. This provides a generalisation of a result by B. Keller, which described the A_infinity-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra. As an application, I will show how to compute the coaugmented curved A_infinity-coalgebra structure of the Koszul codual of a PBW deformation of an N-Koszul algebra, extending a previous result by G. Fløystad and J. Vatne.

Friday, 08 June 2018

• 14:15, Room T2-205
  Mikhail Gorsky (Cologne): Extended Hall algebras
  Abstract: Hall algebras play an important role in representation theory and algebraic geometry. The Hall algebra of an exact or a triangulated category captures information about the extensions between objects. It turns out that in some cases twisted and extended Hall algebras of triangulated categories are well-defined even when their non-extended counterparts are not. I will explain how to associate a twisted extended Hall algebra to a triangulated category, when the latter arises as the homotopy category of a hereditary exact model category or as an orbit category of certain kind. I will discuss applications of this constructions to graded quiver varieties and to categorification of modified quantum group.
• 15:30, Room T2-205
  William Crawley-Boevey (Bielefeld): A new approach to simple modules for preprojective algebras
  Abstract: This is joint work with Andrew Hubery. My earlier work on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an additive analogue of the Deligne-Simpson problem. The last step in the proof of the classification involved some general position arguments; here we give a new approach which avoids such arguments.

Friday, 25 May 2018

• 14:15, Room T2-205
  Sophiane Yahiatene (Bielefeld): Thick subcategories in hereditary abelian categories
  Abstract: Let H be a connected Ext-finite hereditary abelian k-category with tilting complex. In this talk I present a group-theoretic method to classify the thick subcategories generated by exceptional sequences. For that, we consider the Grothendieck group of H, which can be seen as a root lattice of a generalized root system and define a reflection group acting on it.
  As an example we consider the category of coherent sheaves of a weighted projective line of tubular type.
  (Joint work with B. Baumeister and P. Wegener)
  
• 15:30, Room T2-205
  Henning Krause (Bielefeld): Morphic enrichments of triangulated categories (after Keller)
  Abstract: The talk presents some recent work of Bernhard Keller. The notion of a triangulated category suffers from the fact that cones are not functorial. Morphic enrichments provide a concept to overcome this problem, and basically all triangulated categories that arise in nature admit such an enrichment. A morphic enrichment of a triangulated category T is given by a recollement of triangulated categories with T at both ends, and the additive structure of this recollement determines the triangulated structure of T. This idea goes back to work of Keller (Derived categories and universal problems, Comm. Algebra 19, 1991); it turns out to be useful for defining a triangulated structure on the completion of a triangulated category.

Friday, 04 May 2018

• 13:15, Room T2-205
  Magnus Bakke Botnan (München): Representation Theory in Topological Data Analysis
  Abstract: Topological data analysis (TDA) is a relatively recent approach to data analysis in which topological signatures are assigned to data. In this talk I will survey how the theoretical foundations of TDA rest on classical results from the representation theory of quivers. I will also discuss recent results in representation theory inspired by questions in TDA. This is joint work with Ulrich Bauer, Steffen Oppermann and Johan Steen.
• 14:30, Room T2-205
  Haydee Lindo (Stuttgart): Trace modules, Rigidity and Endomorphism rings
  Abstract: I will speak on some recent developments in the theory of trace modules over commutative Noetherian rings. This will include applications of trace modules in understanding endomorphism rings and a discussion of ongoing work examining the relationship between trace modules and modules having no self-extensions.

Friday, 27 April 2018

• 14:15, Room T2-205
  Thomas Poguntke (Bonn): Higher Segal structures in algebraic K-theory and Hall algebras
  Abstract: One of the main results of Dyckerhoff-Kapranov's work on higher Segal spaces concerns the fibrancy properties of Waldhausen's simplicial construction of the algebraic K-theory of an exact category, which are in particular responsible for the associativity of various Hall algebras. We will explain their results, with an emphasis on this latter aspect. Finally, we will introduce a higher dimensional analogue of the construction, where short exact sequences are replaced by longer extensions, whose algebraic ramifications are yet to be clearly understood.
  

Saturday, 21 April 2018

• Workshop "Discrete Categories in Representation Theory"

Friday, 20 April 2018

• Workshop "Discrete Categories in Representation Theory"

Friday, 13 April 2018

• 14:15, Room T2-205
  Minghui Zhao (Beijing): On purity theorem of Lusztig's perverse sheaves
  Abstract: Let Q be a finite quiver without loops and U the quantum group corresponding to Q. Lusztig introduced the canonical basis of the positive part of U via some semisimple perverse sheaves (Lusztig's perverse sheaf). When Q is a Dynkin quiver, Lusztig proved that any Lusztig's perverse sheaf L possesses a Weil structure such that the Frobenius eigenvalues on the stalk of the i-th cohomology sheaf H^i(L) at x are equal to q^(i/2) for any k-rational point x, where k is the finite field of q elements. The purpose of this talk is to generalize this result to all finite quiver without loops. As an application, we shall prove the existence of a class of Hall polynomials. This is a joint work with Jie Xiao and Fan Xu.

Friday, 02 February 2018

• 14:15, Room T2-213
  Gabriel Valenzuela Vasquez (Columbus): Stratification for homotopical groups
  Abstract: The notion of a homotopical group captures and generalizes the properties of compact Lie groups that can be studied using homotopy theory. The goal of this talk is to present a stratification result for the category of modules over the ring spectrum of cochains on G for a broad class of homotopical groups G. We will focus on the techniques inspired by well-established results such as generalizations of Quillen's F-isomorphism theorem, Quillen's stratification theorem, and Chouinard's theorem to the context of homotopical groups. This is joint work with Natalia Castellana, Tobias Barthel, and Drew Heard.
• 15:30, Room T2-213
  Tim Römer (Osnabrück): Commutative algebra up to symmetry and FI-modules
  Abstract: Ideal theory over a polynomial ring in countably many variables is rather complicated. In particular, motivated by results from algebraic statistics and representation theory, one is interested in ideals in such a ring which are invariant under the action of a symmetric group. These kind of ideals can be described by associated ascending chains of symmetric ideals in finitely many variables. In this talk we discuss some new results and open questions of ideals in such chains and their limits. Our approach is based on FI-modules with varying coefficients and various related techniques. This talk is based on joint work with Uwe Nagel.

Friday, 19 January 2018

• 14:15, Room T2-213
  Chrysostomos Psaroudakis (Stuttgart): Reduction techniques for the finitistic dimension
  Abstract: One of the longstanding open problems in Representation Theory of Finite Dimensional Algebras is the so called "Finitistic Dimension Conjecture". The latter homological conjecture is known to be related with other important problems concerning the homological behaviour and the structure theory of finite dimensional algebras. Our aim in this talk is to present some reduction techniques for the finitistic dimension. In particular, we will show that we can remove some vertices and some arrows from a quotient of a path algebra such that the problem of computing the finitistic dimension can be reduced to a possible simpler ("homologically compact") algebra. The results will be illustrated with examples. This is joint work with Edward L. Green and Øyvind Solberg.
• 15:30, Room T2-213
  Victoria Hoskins (Berlin): Group actions on quiver varieties and applications
  Abstract: We study two types of actions on King's moduli spaces of quiver representations over a field k, and we decompose their fixed loci using group cohomology in order to give modular interpretations of the components. The first type of action arises by considering finite groups of quiver automorphisms. The second is the absolute Galois group of a perfect field k acting on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points, which we decompose using the Brauer group of k, and we describe the rational points as quiver representations over central division algebras over k. Over the field of complex numbers, we describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties using the language of branes. This is joint work with Florent Schaffhauser.

Friday, 15 December 2017

• 13:15, Room T2-213
  Julian Külshammer (Stuttgart): Pro-species of algebras and monomorphism categories
  Abstract: Inspired by work of Geiss, Leclerc, and Schroer on geometric realisation of quantum groups of non-simply laced Dynkin diagrams over algebraically closed fields, we introduce the notion of a pro-species of algebras. This concept generalises the concept of a species over a non-algebraically closed field studied intensively by Dlab and Ringel. In a second part we report on joint work in progress with Nan Gao, Sondre Kvamme, and Chrysostomos Psaroudakis on studying monomorphism categories over these pro-species generalising results due to Ringel and Schmidmeier.
• 14:30, Room T2-213
  Oliver Lorscheid (Rio de Janeiro): Representation type via quiver Grassmannians
  Abstract: The representation type of a quiver Q can be characterized by the geometric properties of the associated quiver Grassmannians: (a) Q is representation finite if all quiver Grassmannians are smooth and have cell decompositions into affine spaces; (b) Q is tame if all quiver Grassmannians have cell decompositions into affine spaces and if there exist singular quiver Grassmannians; (c) Q is wild if every projective variety occurs as a quiver Grassmannian.
  
  With the exception of (extended) Dynkin type E, this result is proven in joint work with Thorsten Weist, based on previous results by Haupt, Reineke, Hille and Ringel. The result for type E is work in progress by Cerulli Irelli-Esposito-Franzen-Reineke. In this talk, we will explain this result and parts of its proof.
• 16:00, Room T2-213
  Daniel Kaplan (London): Two perspectives on generalized preprojective algebras
  Abstract: Geiss-Leclerc-Schröer have a series of beautiful papers on generalized preprojective algebras, where truncated polynomial algebras are assigned to each vertex and bimodules to each arrow of a quiver. Geuenich observed that their construction works in the setting of symmetric Frobenius algebras at the vertices. In this talk, I'll introduce two frameworks to think about such algebras: (1) in terms of the zero fiber of an appropriate moment map and (2) as degenerations of ordinary preprojective algebras. The latter allows one to compute the graded Hilbert series, as I'll demonstrate in examples.

Friday, 08 December 2017

• 13:15, Room T2-213
  Michael Wong (Austin): Hochschild cohomology of noncommutative matrix factorizations
  Abstract: R. Bocklandt proved a version of mirror symmetry in which the dual to a punctured curve is a noncommutative Landau-Ginzburg (LG) model: namely, the Jacobi algebra of a dimer model, equipped with a canonical potential. We will review the basic theory of dimer models and existing literature on matrix factorizations of commutative LG models. Then we will present progress towards computing the Hochschild cohomology of matrix factorizations of noncommutative LG models in terms of a compactly supported version for curved algebras.
• 14:30, Room T2-213
  Yuta Kimura (Nagoya): Tilting objects for preprojective algebras associated to Coxeter groups
  Abstract: Let Q be a finite acyclic quiver and w be an element of the Coxeter group of Q.
  Buan-Iyama-Reiten-Scott constructed and studied a 2-Calabi-Yau triangulated category E(w) with cluster tilting objects.
  Amiot-Reiten-Todorov showed that E(w) is triangle equivalent to the cluster category of an algebra A_w.
  In this talk, we consider a triangulated category E(w)^Z which is the Z-graded version of E(w).
  We show that E(w)^Z always has a silting object and give a sufficient condition on w such that the silting object is a tilting object.
  In particular, E(w)^Z is triangle equivalent to the derived category of A_w.
• 16:00, Room T2-213
  Yuriy Drozd (Kiev): Nodal curves, skewed gentle algebras and their maternal envelopes
  Abstract: Nodal curves are introduced and their categorical resolutions are constructed. In rational case a tilting complex is constructed which relates them to skewed gentle algebras.

Friday, 01 December 2017

• 15:15, Room T2-213
  Mehdi Aaghabali (Edinburgh): Graded structure of Leavitt path algebras
  Abstract: One can construct path algebras starting from a graph subject to the relation that if one can not move from one edge along to another one, the product of these edges is zero. So, the nonzero elements of this algebra are all the (finite) paths in the graph. This justifies the name path algebras. One of the main problems in the area is classification of LPAs in terms of an invariant that can be easily calculated from the underlying graph. In this talk we show there are isomorphic LPAs associated to different graphs, however when grading is considered have completely different structures. This indicates that if there is a chance of having a complete invariant for LPAs, that invariant should take into account the grading structure.

Friday, 24 November 2017

• 14:15, Room T2-213
  Michael Ehrig (Sydney): The good old Brauer algebra from a modern view
  Abstract: In the talk I will discuss the Brauer algebra. Starting at its origin in classical invariant theory and then outlining how to link it to a more modern point of view, which includes geometry of perverse sheaves, category O for certain Lie algebras as well as topologically defined Khovanov algebras. This will give a graded presentation of the Brauer algebra and will have applications for orthosymplectic Lie super algebras.

Friday, 17 November 2017

• 14:15, Room T2-213
  Steffen Oppermann (Trondheim): Change of rings and singularity categories
  Abstract: This talk is based on joint work with Chrysostomos Psaroudakis and Torkil Utvik Stai.
  The singularity category of a (finite dimensional) algebra is defined to be the localization of the bounded derived category modulo the subcategory of perfect complexes. The name "singularity category" is motivated by commutative algebra, where the singularity category contains information about the singularities of a ring while forgetting the regular parts. For (non-commutative) finite dimensional algebras the meaning is less clear.
  The aim of my talk is to investigate when ring-morphisms induce functors between singularity categories (and related cocomplete categories). One may hope that this gives some idea what information survives in the singularity category.
  

Thursday, 09 November 2017

• BiBo Seminar
  12:15, Room V5-227
  Magdalena Boos (Bochum): The algebraic U-Quotient of the nilpotent cone
  Abstract: We consider the conjugation-action of the standard unipotent subgroup U of GL_n(C) on the nilpotent cone N of complex nilpotent matrices of square-size n. The structure of the invariant ring C[N]^U (and, thus, the algebraic quotient X:=Spec C[N]^U) is not known yet. In this talk, we discuss a generic normal form of the U-orbits in N, define a set of U-invariants which span C[N]^U and use these concepts to generically separate the U-orbits. This is work in progress and we end the talk by discussing different ideas to approach the explicit structure of C[N]^U. (Joint with H. Franzen and M. Reineke)
• BiBo Seminar
  13:45, Room V5-227
  Andrew Hubery (Bielefeld): Preprojective algebras revisited

Friday, 03 November 2017

• 14:15, Room T2-213
  Dirk Kussin (Paderborn): What is a tube?
  Abstract: We discuss the categorical structure of a tube, let say a homogeneous one over a tame hereditary algebra over a field (or more generally, over a noncommutative regular projective curve), and compare a bottom-up with a top-down approach for its determination. We compare the functorial properties of the Auslander-Reiten translation on a tube with tubular shift functors associated with tubes. Some new results and examples will be presented.

Friday, 20 October 2017

• 14:15, Room T2-213
  Jeanne Scott (Bogotá): Towards a Jucys-Murphy theory for the Okada algebras
  Abstract: I'll discuss work in progress which aims to construct Jucys-Murphy elements in the Okada algebra F_n together with a corresponding notion of content for the Young-Fibonacci lattice which encodes the spectrum of the Jucys-Murphy elements with respect to the Fibonacci-Tableau bases for irreducible F_n-representations.

Friday, 13 October 2017

• 14:15, Room T2-213
  Alexander Samokhin (Düsseldorf): T-structures on the derived categories of coherent sheaves on flag varieties and the Frobenius morphism
  Abstract: We will talk about semiorthogonal decompositions of the derived categories of coherent sheaves on flag varieties that are compatible with the action of Frobenius morphism on coherent sheaves via push-forward and pull-back functors. We start with an example of such a decomposition, and, in particular, show how it implies Kempf's vanishing theorem. In some cases, refinements of that decomposition define, via derived Morita equivalence, the non-standard t-structures on the derived categories of flag varieties. These t-structures and their duals are related to each other via an autoequivalence of the ambient derived category whose square is isomorphic to the Serre functor. We will treat in detail the case of the groups of rank two.

Thursday, 14 September 2017

• 13:15, Room V2-200
  David Ploog (Berlin): Exact tilting theory
  Abstract: Tilting theory has proved very important in algebraic geometry and representation theory, for the construction of autoequivalences and for linking varieties and algebras. In joint work with Lutz Hille, we describe a geometric setup, where the tilting equivalence is exceptionally strong: it restricts to an equivalence of abelian categories. In this talk, we explain the categorical background and the geometric side.
  
• 14:15, Room V2-200
  Patrick Wegener (Bielefeld): Braid group action in elliptic Weyl groups and classification of thick subcategories
  Abstract: In 2010 Igusa, Schiffler and Thomas classified the set of thick subcategories of the bounded derived category of mod(A) generated by an exceptional sequence, where A is a hereditary Artin algebra, in terms of the poset of noncrossing partitions. Motivated by a Theorem of Happel we consider the category of coherent sheaves on a weighted projective line of tubular type instead of mod(A) and give an outlook how to obtain a similar classification for this case. As an important tool we show that the braid group acts transitively on factorizations of the Coxeter element in an elliptic Weyl group of tubular type.
  
• 15:45, Room V2-200
  David Ploog (Berlin): Exceptional sequences for the Auslander algebra of the fat point
  Abstract: This algebra is well-known in representation-theory. We classify spherical modules and full exceptional sequences over this algebra. There are combinatorial left/right symmetric group actions on these sequences. We categorify both of these, using spherical twists and right mutations. All of this can be nicely visualised using worm diagrams.

Friday, 28 July 2017

• 13:15, Lecture Hall H10
  Xiao-Wu Chen (Hefei): The dual group actions and stable tilting objects
  Abstract: Weighted projective lines of different tubular types are related via the equivariantization with respect to certain cyclic group actions. It induces a bijection between the classification of tau^2-stable tilting sheaves and the one of g-stable tilting sheaves for some automorphism g on the weighted projective lines. The bijection holds in the general setting for the dual group actions on triangulated categories. This is joint with Jianmin Chen and Shiquan Ruan.
• 14:45, Lecture Hall H10
  Hideto Asashiba (Shizuoka): Cohen-Montgomery duality of bimodules with applications to equivalences of Morita type
  Abstract: We fix a group G and a commutative ring k, and assume that all categories in consideration are skeletally small k-categories with projective Hom-spaces. Let R, S be categories with G-actions and A, B G-graded categories such that there exist G-covering functors R —> A and S —> B. Few years ago we established an equivalence between G-invariant S-R-bimodules and G-graded B-A-bimodules, an analogue of the so-called Cohen-Montgomery duality, and as an application gave one-one correspondence between G-invariant stable equivalence of Morita type between R and S and G-graded stable equivalence of Morita type between A and B. This is extended to also standard derived equivalences and singular equivalences of Morita type.

Friday, 21 July 2017

• 15:30, Lecture Hall H10
  Klaus Bongartz (Wuppertal): Representation embeddings and the second Brauer-Thrall conjecture
  Abstract: We prove that there is a representation embedding from the category of finite dimensional representations of the Kronecker quiver without simple injective direct summand to the category of finite dimensional A-modules as soon as A is a representation-infinite finite-dimensional algebra over an algebraically closed field. This is in a sense the strongest possible version of the second Brauer-Thrall conjecture and the proof is independent of Drozd's theorem.
  In the talk I also sketch the history of some more results connected with the Brauer-Thrall conjectures and essential steps in their proof as I understand them.

Friday, 14 July 2017

• 13:15, Lecture Hall H10
  Sondre Kvamme (Bonn): On Marczinzik's observations regarding the Nakayama conjecture
  Abstract: About a year ago Rene Marczinzik wrote a short note where he shows that the Nakayama conjecture is implied by a statement made by Beligiannis in Lemma 6.19 part (3) in "The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stabilization". However, there is no known proof of Beligiannis' statement. In this talk I will state this lemma and show how it implies the Nakayama conjecture. Also, I will explain how a counterexample to this lemma gives a counterexample to the generalized Nakayama conjecture.
• 14:30, Lecture Hall H10
  Rasool Hafezi (Isfahan): On finitely presented functors over the stable categories
  Abstract: In this talk, I will explain my recent work available on arXiv with same title as my talk here.
  In this paper, I studied the category of finitely presented functor over the stable category of some certain subcategories of an abelian category. In particular, this investigation provides a positive answer to a conjecture of M. Auslander, that is, a direct summand of a covariant Ext-functor is again of that form. I will continue this study for the subcategory of Gorenstein projective modules, and as a result this gives some criteria when the relative Auslander translation respect to this subcategory is the first syzygy functor.
  
• 16:00, Lecture Hall H10
  Xiao-Wu Chen (Hefei): Introducing K-standard additive categories
  Abstract: We introduce the notion of K-standard additive category. This is motivated by the following open question of Jeremy Rickard: is any derived equivalence standard? We will report some progress on this question and the related ones.

Friday, 30 June 2017

• 14:15, Lecture Hall H10
  Otto Kerner (Duesseldorf): Thick subcategories of the stable category of modules over the exterior algebra
  Abstract: Let R be a finite dimensional exterior algebra over an algebraically closed field. R is a graded algebra in the obvious way. We consider the graded category of finite dimensional R-modules (with homomorphisms of degree zero). The corresponding stable category is a triangulated category is equivalent as a triangulated category to the derived category of coherent sheaves over the corresponding projective space). We describe the thick subcategories of this category, generated by R-modules of complexity one.
  This is joint work with Dan Zacharia.

Friday, 23 June 2017

• Bielefeld - Bochum Seminar
  It takes place in Bochum.

Friday, 09 June 2017

• 14:15, Lecture Hall H10
  Julia Sauter (Bielefeld): Desingularizations of orbit closures and quiver Grassmannians from tilting modules
  Abstract: We study orbit closures in representation spaces of finite-dimensional algebras and quiver Grassmannians. In both cases we construct desingularizations assuming the module is gen-finite, i.e. it has only finitely many isomorphism classes of quotients. Our construction uses tilting modules on endomorphism rings of generators, recollements and homotopy categories. This is joint work in progress with Matthew Pressland generalizing previous work from Sauter--Crawley-Boevey.
• 15:30, Lecture Hall H10
  Tobias Rossmann (Auckland): The average size of the kernel of a matrix and orbits of linear groups
  Abstract: Motivated by questions on the representation growth of unipotent algebraic groups, we study generating functions enumerating orbits of p-adic linear groups. Using Lie theory, these functions turn out to be closely related to average sizes of kernels in modules of matrices.

Friday, 02 June 2017

• 13:15, Lecture Hall H10
  Apolonia Gottwald (Bielefeld): Lattices of subobject closed subcategories
  Abstract: For an Abelian length category A, we consider the lattice S(A) of full additive subobject closed subcategories. This lattice is distributive if and only if certain conditions on the Ext-quiver of A are fulfilled. If the Ext-quiver is symmetric, then S(A) is distributive if and only if an even stronger property holds: every subobject of an indecomposable object in A is itself indecomposable. In particular, we get this equivalence, if A is the category of finitely generated modules over an Artin algebra over an algebraically closed field.
• 14:30, Lecture Hall H10
  Alex Martsinkovsky (Boston): (Co-)torsion via stable functors
  Abstract: This talk will concentrate on two new applications of stable functors (these are functors defined on injectively or projectively stable module categories). The first one is a definition of the torsion submodule of a module, which provides a simultaneous generalization of the classical torsion and, for finitely presented modules, of the 1-torsion (= the kernel of the bidualization map). The second one is a definition of the cotorsion quotient module of a module, which doesn’t seem to have a classical prototype. This is done in utmost generality: for arbitrary modules over arbitrary rings. Some of the obtained results are new even in the classical setting of abelian groups.
  The new definitions are remarkably simple and can be given without appealing to stable functors. However, one of the goals of this talk is to convince the audience that the language of functors, being simple, convenient, and natural, brings about additional insights. In that language, this talk is about the injective stabilization of the tensor product and the projective stabilization of the contravariant Hom functor.
  Time permitting, we shall see that the Auslander-Gruson-Jensen functor sends the cotorsion functor to the torsion functor. If the injective envelope of the ring is finitely presented, then the right adjoint of the AGJ functor sends the torsion functor back to the cotorsion functor. In particular, over an artin algebra, this correspondence establishes a duality between the requisite functors on the categories of all modules.
  This will be an expository talk, no prior familiarity with functor categories is assumed. This is joint work with Jeremy Russell.
• 16:00, Lecture Hall H10
  Dan Zacharia (Syracuse): Using linear modules to study exceptional sheaves on the projective n-space
  Abstract: I will talk on joint work with Otto Kerner. Let V be an n+1 dimensional vector space over the field of complex numbers, let R be the exterior algebra on V, and let S be the polynomial algebra in n+1 indeterminates. Finally, consider the category of coherent sheaves on the corresponding projective space. A coherent sheaf E is called exceptional, if it has no extensions with itself, and, in addition, E has an endomorphism ring isomorphic to the ground field. My talk will be about a possible way to reduce certain problems about coherent sheaves (in particular, exceptional ones) to working with some particularly nice modules (called linear modules) over the exterior algebra.

Friday, 26 May 2017

• 13:15, Lecture Hall H10
  Estanislao Herscovich (Grenoble): On some mixture conditions of monoidal structures appearing in Quantum Field Theory
  Abstract: R. Borcherds has introduced a different point of view to formalise perturbative Quantum Field Theory (pQFT). In particular, he uses several objects which behave somehow like bialgebras and comodules over them, and which are essential in his definition of Feynman measure. The former objects don’t seem however to be bialgebras in the classical sense, for their product and coproduct are with respect to two different tensor products, and similarly for comodules. Moreover, following physical motivations, these objects are given as some symmetric constructions of geometric nature.
  The aim of this talk is on the one hand to show that the “bialgebras” and “comodules” introduced by Borcherds cannot “naturally” exist, and on the other side to provide a background where a modified version of the so-called “bialgebras” and “comodules” do exist. This involves a category provided with two monoidal structures satisfying some compatibility conditions. As expected, the modified version of the mentioned “bialgebras” and “comodules” are not so far from the original one, considered by Borcherds. Moreover, we remark that these new candidates allowed us to prove the main results stated by Borcherds in his article (see my manuscript "Renormalization in Quantum Field Theory").
• 14:30, Lecture Hall H10
  Dave Benson (Aberdeen): Surface bundles over surfaces, and cohomology of finite groups
  Abstract: This is a report on joint work with Caterina Campagnolo, Andrew Ranicki and Carmen Rovi. The signature of a surface bundle over a surface is always divisible by four. We describe how to compute the signature modulo eight using the cohomology and representation theory of finite groups.
• 16:00, Lecture Hall H10
  Jon Carlson (Athens, Georgia): An obstreperous class of modules
  Abstract: We discuss several questions concerning the nature of modules over a modular group algebra having full support variety but which have dimension divisible by the prime characteristic of the coefficient field. We will mention some results and work with Paul Balmer and with Dave Benson.

Friday, 05 May 2017

• 13:15, Lecture Hall H10
  Ming Lu (Chengdu): Modified Ringel-Hall algebras and Drinfeld double
  Abstract: Inspired by the works of Bridgeland and Gorsky, we define an algebra from the Ringel-Hall algebra of the category formed by Z/2-graded complexes over a hereditary abelian category which may not have enough projective objects. Such algebra is called modified Ringel-Hall algebra. We prove it to be of some nice properties and structures. The first one is that it has a nice basis, which yields that it is a free module over a suitably defined quantum torus of acyclic complexes, with a basis given by the isomorphism classes of objects in the derived category of Z/2-graded complexes. The second one is that in twisted case it is isomorphic to the Drinfeld double Ringel-Hall algebra of the hereditary abelian category. Finally, if the category has a tilting object T, then its modified Ringel-Hall algebra is isomorphic to the Z/2-graded semi-derived Hall algebra and also the Bridgeland's Ringel-Hall algebra of the endomorphism algebra of T.
  This is a joint work with Liangang Peng, and it is available at arXiv:1608.03106.
• 14:30, Lecture Hall H10
  Jorge Vitória (London): Silting and cosilting classes in derived categories
  Abstract: A class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Cotilting classes, on the other hand, are precisely the resolving and definable subcategories of the module category whose Ext-orthogonal class has bounded injective dimension.
  Silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules. In this talk we will discuss a generalisation of the characterisations above to this derived setting, with a particular focus on the silting case. This is joint work with Frederik Marks.
• 16:00, Lecture Hall H10
  Olaf Schnürer (Bonn): Geometric applications of conservative descent for semi-orthogonal decompositions
  Abstract: Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry we introduce a technique called "conservative descent” in order to establish such decompositions locally. The decompositions we have in mind are those for projective bundles, blow-ups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semi-orthogonal decompositions for Brauer-Severi varieties.
  This is joint work in progress with Daniel Bergh.

Friday, 28 April 2017

• 14:15, Lecture Hall H10
  Chun-Ju Lai (Bonn): Affine Hecke algebras and quantum symmetric pairs
  Abstract: In an influential work of Beilinson, Lusztig and MacPherson, they provide a construction for (idempotented) quantum groups of type A together with its canonical basis. Although a geometric method via partial flags and dimension counting is applied, it can also be approached using Hecke algebras and combinatorics. In this talk I will focus on the Hecke algebraic approach and present our work on a generalization to affine type C, which produces favorable bases for q Schur algebras and certain coideal subalgebras of quantum groups of affine type A. We further show that these algebras are examples of quantum symmetric pairs, which are quantization of symmetric pairs consisting of a Lie algebra and its fixed-point subalgebra associated to an involution.
  This is a joint work (arXiv:1609.06199) with Z. Fan, Y. Li, L. Luo, and W. Wang.
• 15:30, Lecture Hall H10
  Charles Vial (Bielefeld): Numerical obstructions to the existence of exceptional collections on surfaces
  Abstract: I will give a complete classification of smooth projective complex surfaces that admit a numerically exceptional collection of maximal length. I will also give arithmetic constraints for the existence of such collections on surfaces defined over non-algebraically closed field.

Friday, 21 April 2017

• 13:15, Lecture Hall H10
  Pieter Belmans (Antwerpen): Hochschild cohomology of noncommutative planes and quadrics
  Abstract: The derived category of P^2 has a full exceptional collection, described by the famous Beilinson quiver and its relations. One can compute the Hochschild cohomology of P^2 as the Hochschild cohomology of this finite-dimensional algebra, for which many tools are available. Changing the relations in the quiver corresponds to considering derived categories of noncommutative planes.
  I will explain how these (and the noncommutative analogues of the quadric surface) are described using Artin-Schelter regular (Z-)algebras, and how one can use their classification to compute the Hochschild cohomology of all finite-dimensional algebras obtained in this way, exhibiting an interesting dimension drop.
  If time permits I will explain how it is expected that the fully faithful functor between the derived category of P^2 and its Hilbert scheme of two points relates their Hochschild cohomologies, and how this is expected to behave under deformation.
• 14:30, Lecture Hall H10
  Shiquan Ruan (Beijing): Applications of mutations in the derived categories of weighted projective lines to Lie and quantum algebras
  Abstract: Let coh-X be the category of coherent sheaves over a weighted projective line X and let D^b(coh-X) be its bounded derived category. In this talk we will focus on the study of the right and left mutation functors arising in D^b(coh-X) attached to certain line bundles. We first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver Q associated with X. By further dealing with the Ringel–Hall algebra of X, we show that these functors provide a realization for Tits’ automorphisms of the Kac–Moody algebra g_Q of Q, as well as for Lusztig’s symmetries of the quantum enveloping algebra of g_Q.
• 16:00, Lecture Hall H10
  Thorsten Weist (Wuppertal): Normal forms for quiver representations induced by tree modules
  Abstract: With a fixed tree module of a quiver and a fixed tree-shaped basis of its group of self-extensions, one can associate an affine space of representations with the same dimension vector. If all representations in this cell are indecomposable and pairwise non-isomorphic, this induces a normal form for these representations. Even if this is general not true, it is possible to state conditions which are sufficient to obtain cells of pairwise non-isomorphic indecomposable representations. For instance this is the case if the tree module is a torus fixed point under a certain torus action. But also recursive constructions of indecomposable representations turn out to be very useful. Now it also might happen that the same representation lies in different cells. Nevertheless, for certain roots this machinery can be used to give a full description of the set of indecomposable representations. In any case, this gives a concept which can be used to deduce normal forms for indecomposable representations of quivers based on the notion of tree modules.
  We also discuss connections to a rather vague conjecture of Kac which says that the set of isomorphism classes of indecomposable representations of quivers admits a cell decomposition into locally closed subvarieties which are affine cells.

Friday, 07 April 2017

• 14:15, Room U2-113
  Robert Marsh (Leeds): Rigid and Schurian modules over tame cluster-tilted algebras
  Abstract: We classify the indecomposable rigid and Schurian modules over a cluster-tilted algebra of tame representation type. Such a cluster-tilted algebra B has an associated cluster algebra A(Q), where Q is the quiver of the algebra. Answering a question of T. Nakanishi, we show that A(Q) can have a denominator vector which is not the dimension vector of any indecomposable B-module. Using the above classification, we show that every denominator vector of A(Q) is the sum of the dimension vectors of at most three indecomposable rigid B-modules.
  The talk is based on joint work with Idun Reiten.

Thursday, 09 March 2017

• 16:15, Room U2-135
  Michael Wemyss (Glasgow): Faithful actions in algebra and geometry
  Abstract: In many algebraic and geometric contexts, the associated derived category admits an action by the fundamental group of some reasonable topological space. In other words, there is a group homomorphism from the fundamental group to the group of autoequivalences. The most famous examples are actions by braid groups, but there are many more general examples, including actions induced by 3-fold flops in algebraic geometry.
  
  I will explain one technique, based on exploiting the partial order on tilting modules, that can be used to deduce when the action is faithful, that is, when the group homomorphism is injective. This algebraic framework applies in various settings, and can be used to extract geometric corollaries, including some in the motivating example of flops. This is joint work with Yuki Hirano.

Tuesday, 28 February 2017

• 14:15, Room V2-213
  Ralf Schiffler (Storrs, Connecticut): Cluster algebras, snake graphs and continued fractions
  Abstract: This talk is on a combinatorial realization of continued fractions in terms of perfect matchings of the so-called snake graphs, which are planar graphs that have first appeared in expansion formulas for the cluster variables in cluster algebras from triangulated marked surfaces. ​I will also explain applications to cluster algebras, as well as​ to​ elementary number theory. This is​ a​ joint work with Ilke Canakci​.​

Friday, 03 February 2017

• 14:15, Room U2-135
  Teresa Conde (Stuttgart): Strongly quasihereditary algebras
  Abstract: Quasihereditary algebras are abundant in mathematics. They classically occur as blocks of the category O and as Schur algebras.
  They also occur as endomorphism algebras associated to modules endowed with special filtrations. The quasihereditary algebras produced in these cases are very often strongly quasihereditary in the sense of Ringel. The ADR algebra of a finite-dimensional algebra A is an example of such an algebra. Other examples of strongly quasihereditary algebras include: the Auslander algebras; the endomorphism algebras constructed by Iyama, used in his proof of the finiteness of the representation dimension; certain cluster-tilted algebras studied by Geiß-Leclerc–Schröer and Iyama–Reiten.
  In this talk I will start by introducing the ADR algebra and by describing its neat quasihereditary structure. I will then look at larger classes of strongly quasihereditary algebras and describe some of their properties.

Friday, 20 January 2017

• 14:15, Room U2-135
  Daniel Bissinger (Kiel): Invariants of regular components for wild Kronecker algebras
  Abstract: Motivated by the work on modular representation theory of finite group schemes, Worch introduced the categories of modules with the equal images property and the equal kernels property for the generalized Kronecker algebra.
  Given a regular component C of the Auslander-Reiten quiver, we study the distance W(C) between the two non-intersecting cones in C given by modules with the equal images and the equal kernels property.
  We show that W(C) is closely related to the quasi-rank rk(C) of C. Utilizing covering theory, we discuss how to construct for each natural number n a regular component C_n with W(C_n) = n.
• 15:30, Room U2-135
  Kevin De Laet (Antwerpen): The connection between Sklyanin algebras and the finite Heisenberg groups
  Abstract: The 3-dimensional quadratic Sklyanin algebras are noncommutative graded analogues of the polynomial ring in 3 variables and have been studied by people like Artin, Tate, Van den Bergh, ... In this talk, I am going to show how these algebras can be constructed using the representation theory of the finite Heisenberg group of order 27 such that this group acts on these algebras as gradation preserving automorphisms.
  This action will then be used to prove certain results regarding the central element of degree 3 of such algebras. This talk is based on my paper https://arxiv.org/abs/1612.06158

Friday, 13 January 2017

• 14:15, Room C01-226
  Liran Shaul (Bielefeld): A well behaved category of derived commutative rings over a noetherian ring
  Abstract: Given a commutative noetherian ring K, the goal of this talk is to present a category of derived commutative rings over K which includes the finite type K-algebras, and is closed under the operations of localization, derived tensor products, and derived adic completion. To do this we introduce the homotopy category of derived commutative rings with an adic topology, and explain how to perform these various operations in this category. In particular, we construct the derived adic completion of a derived commutative ring A with respect to a finitely generated ideal of the ring H^0(A).

Friday, 16 December 2016

• 13:15, Room U2-135
  Ögmundur Eiriksson (Bielefeld): From submodule categories to the stable Auslander algebra
  Abstract: C. Ringel and P. Zhang have studied a pair of functors from the submodule category of a truncated polynomial ring over a field to a preprojective algebra of type A. We present the analogous process starting with any self-injective algebra of finite representation type over a field k.
  To this end we study two functors from the submodule category to the module category of the stable Auslander algebra. The functors are compositions of objective functors, and both factor through the module category of the Auslander algebra. We are able to describe the kernels of these functors, both of which have finitely many indecomposables.
  One of the functors factors through the subcategory of torsionless modules over the Auslander algebra. That subcategory arises as the subcategory of objects with a filtration by standard modules for a quasi-hereditary structure on the Auslander algebra if and only if our original algebra is uniserial.
• 14:30, Room U2-135
  Florian Gellert (Bielefeld): Maximum antichains in subrepresentation posets
  Abstract: For indecomposable representations of Dynkin quivers, the structure of indecomposable morphisms is given by the Auslander-Reiten quiver. Very different posets can be formed if one considers the (not necessarily indecomposable) monomorphisms alone; the poset by inclusion admits particular properties. In this talk we study the latter poset for various orientations of type A quivers. We construct maximum antichains and obtain formulas for the widths of the respective posets. This is joint work with Philipp Lampe.
• 16:00, Room U2-135
  Rene Marczinzik (Stuttgart): On dominant dimensions of algebras
  Abstract: The famous Nakayama conjecture states that every nonselfinjective finite dimensional algebra has finite dominant dimension. A stronger conjecture is the conjecture of Yamagata: The dominant dimension of non-selfinjective algebras with a fixed number s of simple modules is bounded by a finite function depending on s. We show that Yamagata's conjecture is true for monomial algebras with n simple modules. In fact an explicit optimal bound is given by 2n-2, which answers a conjecture of Abrar who conjectured bounds in the special case of Nakayama algebras. We end the talk with several conjectures and open questions related to dominant dimension. Those conjectures are mainly motivated by computer calculations with the GAP package QPA. The conjectures are related to a new construction of higher Auslander algebras and periodic algebras from representation-finite hereditary algebras and a new interplay between homological dimensions and the combinatorics of Dyck paths and circular codes for special classes of algebras.

Friday, 02 December 2016

• NWDR Workshop Winter 2016
  10:30, Room V2-210/216
  Gabor Elek (Lancaster): Convergence and limits of finite dimensional representations of algebras
  Abstract: Motivated by the limit theory of finite graphs I will introduce the notion of metric convergence of finite dimensional representations of algebras over a countable field. It turns out that the limit points are infinite dimensional representations and together with the finite dimensional representations they form a compact metric space. I will also talk about the notion of hyperfiniteness for finite dimensional algebras and its relation with the classical notion of amenability.
  
• NWDR Workshop Winter 2016
  11:45, Room V2-210/216
  Andrew Hubery (Bielefeld): Euler characteristics of quiver Grassmannians
  Abstract: We show how to define an Euler characteristic for the Grassmannian of submodules of a rigid module, for any finite-dimensional hereditary algebra over a finite field. Moreover, we prove that this number is positive whenever the Grassmannian is non-empty, generalising the known case of quiver Grassmannians. As an application, this shows that, for any symmetrisable acyclic cluster algebra, the Laurent expansion of a cluster variable always has non-negative coefficients.
• NWDR Workshop Winter 2016
  14:00, Room V2-210/216
  Peter Patzt (Berlin): Representation stability for the general linear groups
  Abstract: The notion of representation stability for the symmetric groups, the general linear groups and the symplectic groups was introduced by Church-Farb. We give a criterion for a sequence of algebraic representations of the general linear groups to be representation stable. With it we prove that the factors of the lower central series of the Torelli subgroups of the automorphism groups of free groups are representation stable.
• NWDR Workshop Winter 2016
  15:30, Room V2-210/216
  Pierre-Guy Plamondon (Paris): Multiplication formulas
  Abstract: In the past decade, the study of cluster algebras via representations of quivers has proved a successful way to tackle some of the problems in the theory. In this talk, I will review the theory of cluster characters and present new multiplication formulas relating them.
• NWDR Workshop Winter 2016
  16:45, Room V2-210/216
  Christine Bessenrodt (Hannover): Kronecker products of characters of the symmetric groups and their double covers
  Abstract: Decomposing Kronecker products of irreducible characters of the symmetric groups (or equivalently, of inner products of Schur functions) is a longstanding central problem in representation theory and algebraic combinatorics. The talk will focus on special Kronecker products and related problems for skew characters, in particular on the recent classification of multiplicity-free Kronecker products of irreducible characters of the symmetric groups, conjectured in 1999. Also related conjectures and results on spin characters of the double cover groups will be discussed, and the connection between them will be illustrated by some applications of spin characters towards results for symmetric groups.

Friday, 25 November 2016

• 14:15, Room U2-135
  Matthew Pressland (Bonn): Dominant dimension and canonical tilts
  Abstract: Any finite dimensional algebra with dominant dimension d admits a 'canonical' k-tilting module for each k from 0 to d, each giving a derived equivalence with some algebra B_k. These tilts have very special properties; for example, they never increase the global dimension. In the case of the Auslander algebra of a representation-finite algebra A, Crawley-Boevey and Sauter (generalising Cerulli Irelli, Feigin and Reineke) used the tilt B_1 to construct desingularisations of certain varieties of A-modules. More generally, for d at least 2, any algebra of dominant dimension d is the endomorphism algebra of a generating-cogenerating module M over some algebra A, and many of the results for Auslander algebras have analogues in this setting. In particular, we may realise each B_k as an endomorphism algebra in the homotopy category of A, an observation which we can exploit to describe rank varieties, of arbitrary finite dimensional modules over arbitrary finite dimensional algebras, as affine quotient varieties. We may also use the canonical tilting modules to give a new characterisation of d-Auslander (or, more generally, d-Auslander–Gorenstein) algebras. This is joint work with Julia Sauter.
  

Friday, 18 November 2016

• 14:15, Room U2-135
  Sondre Kvamme (Bonn): A generalization of finite-dimensional Iwanaga-Gorenstein algebras
  Abstract: We will introduce certain well-behaved comonads on abelian categories, which generalize features of the module category of a finite dimensional algebra. For example, we will define Gorenstein flat objects relative to the comonad, which generalize Gorenstein projective modules for a finite-dimensional algebra. We will also define what it means for the comonad to be Gorenstein, and state analogues of some classical results for Iwanaga-Gorenstein algebras. We will illustrate the constructions and results on specific examples.
• 15:30, Room U2-135
  Rosanna Laking (Bonn): Krull-Gabriel dimension, Ziegler spectra of module categories and applications to compactly generated triangulated categories.
  Abstract: We will begin by defining the notion of Krull-Gabriel (KG-) dimension for the module category of a ring R (with many objects) and outlining how this relates to a topology on the indecomposable pure-injective R-modules. Using examples, we aim to explain how the KG-dimension "measures" transfinite factorisations of morphisms in R-mod.
  We will then consider analogous notions for compactly generated triangulated categories. We will show that one can work in the context of an associated module category, and hence one can directly make use of the tools described in the first part of the talk. Using this insight we will describe the Ziegler spectrum of the bounded derived category of a derived-discrete algebra A and calculate its KG-dimension. These techniques lead to a classification of the indecomposable objects in the (unbounded) homotopy category of A.
  This talk is based on joint work with K. Arnesen, D. Pauksztello and M. Prest.

Friday, 04 November 2016

• 14:15, Room U2-135
  Shengfei Geng (Chengdu): Tilting modules and support tau-tilting modules over preprojective algebras associated with symmetrizable generalized Cartan matrices
  Abstract: For each skew-symmetrizable generalized Cartan matrix, Geiss-Leclec-Schröer defined a class of preprojective algebra which concide with the classical preprojective algebra when the Cartan matrix is symmetric and the symmetrizer is an identity matrix. In this paper, we proved that there is a bijection between the sets of cofinite tilting ideals with global dimension at most one of such preprojective algebra and the corresponding Weyl group when the preprojective algebra is non-Dynkin type. Based on this, we proved that there is a bijection between the sets of support tau-tilting modules of the preprojective algebra and the corresponding Weyl group when the preprojective algebra is of Dynkin type. Here the preprojective algebras of Dynkin type contain not only types of A,D,E, but also contain types of B,C,G,F. These results generalized the results over classical preprojective algebras.
• 15:30, Room U2-135
  Ming Lu (Chengdu): Singularity categories of positively graded Gorenstein algebras
  Abstract: This is a report on ongoing work with Bin Zhu. We discuss the existence of silting objects and tilting objects in the singularity categories of graded modules over positively graded Gorenstein algebras. By generalizing a result of Yamaura for positively graded selfinjective algebras, we prove that for a positively graded 1-Gorenstein algebra A such that A_0 has finite global dimension, its singularity category of graded modules has a silting object. Under some conditions, this silting object is even a tilting object. After that, we apply it to cluster-tilted algebras and representations of quivers over local rings.

Friday, 28 October 2016

• 13:15, Room U2-135
  Gustavo Jasso (Bonn): Mesh categories of type A-infinity and tubes in higher Auslander-Reiten theory
  Abstract: This is a report on joint work with Julian Külshammer. We construct higher analogues of mesh categories of type A-infinity and of the tubes from the viewpoint of Iyama's higher Auslander-Reiten theory. Our construction relies on unpublished work by Darpö and Iyama. We relate these constructions to higher Nakayama algebras, which we also introduce.
  
• 14:30, Room U2-135
  Julian Külshammer (Stuttgart): Spherical objects in higher Auslander-Reiten theory
  Abstract: This is a report on ongoing work with Gustavo Jasso. Let D be the triangulated category associated to a spherical object of dimension m
  greater than or equal to 2. By work of Jørgensen, this is an m-Calabi-Yau triangulated category with almost split triangles. Moreover, its Auslander-Reiten quiver has m-1 connected components of type ZA-infinity. Building upon work of Amiot, Guo, Keller, and Oppermann-Thomas, for each positive integer d we construct an md-Calabi-Yau (d+2)-angulated category with almost split (d+2)-angles. Moreover, its higher Auslander-Reiten quiver has m-1 connected components of higher mesh type A-infty. For m=2, our construction is analogous to the cluster a category of type A-infinity introduced by Holm-Jørgensen.
• 16:00, Room U2-135
  Sven-Ake Wegner (Wuppertal): Is functional analysis a special case of tilting theory?
  Abstract: The objects of functional analysis together with the corresponding morphisms don't form abelian categories. Classical examples, e.g., the category of Banach spaces, satisfy almost all axioms of an abelian category but the canonical morphism between the cokernel of the kernel and the kernel of the cokernel of a given map fails in general to be an isomorphism. In 2003, Bondal and van den Bergh showed that there is a correspondence between (co-)tilting torsion pairs and so-called quasiabelian categories. Indeed, every quasiabelian category, in particular the category of Banach spaces, is derived equivalent to the (abelian) heart of the canonical t-structure on its derived category. In this talk we discuss examples of categories appearing in functional analysis that are not quasiabelian but which carry a natural exact structure. The derived category is thus defined. We are interested in an extension of the aforementioned equivalence as this could be a step towards a successful categorification of certain analytic problems.

Friday, 21 October 2016

• 14:15, Room U2-135
  Michael K. Brown (Bonn): Topological K-theory of dg categories of graded matrix factorizations
  Abstract: Topological K-theory of complex-linear dg categories is a notion recently introduced by A. Blanc. The main goal of the talk is to discuss a calculation of the topological K-theory of the dg category of graded matrix factorizations associated to a complex quasi-homogeneous polynomial in terms of a classical topological invariant of a hypersurface: the Milnor fiber and its monodromy. I will also discuss some applications of this calculation, and, if time permits, some future directions.

Friday, 26 August 2016

• 14:15, Room V5-227
  Zhi-Wei Li (Xuzhou): A homotopy theory of additive category with suspensions
  Abstract: We give a definition of partial one-sided triangulated categories. We show that complete cotorsion pairs in exact categories, torsion pairs and mutation pairs in triangulated categories all extend to partial one-sided triangulated categories. We prove that partial one-sided triangulated categories yield one-sided riangulated categories by passing to stable categories. We give three areas of application of this result. The first one is the constructions of stable abelian and exact categories which extend work of Koenig-Zhu, Keller-Reiten and Kussin-Lenzing-Meltzer. The second one is the construction of stable triangulated categories which allows us to model Iyama-Yoshino subfactors of triangulated categories via Quillen closed model structures. The last one is to develop a homotopy theory of additive categories with suspensions via Gabriel-Zisman localization which leads to a Buchweitz type theorem in triangulated categories. This theorem extends the recent work of Wei and Iyama-Yang which are generalizations of Buchweitz's work on singularity categories. As a corollary we give a triangle equivalence between Verdier quotients and Iyama-Yoshino subfactors of triangulated categories under suitable conditions.

Thursday, 21 July 2016

• 16:15, Room V3-204
  Anna Felikson (Durham): Geometric realizations of quiver mutations
  Abstract: Mutations of quivers are simple combinatorial transformations introduced in the context of cluster algebras, they appear (sometimes completely unexpectedly) in various domains of mathematics and physics. In this talk we discuss connections of quiver mutations with reflection groups acting on vector spaces and with groups generated by point symmetries in the hyperbolic plane. We show that any mutation class of rank 3 quivers admits a geometric presentation via such a group and that the properties of this presentation are controlled by the Markov constant p^2+q^2+r^2-⁠pqr, where p,q,r are the weights of the arrows in the quiver. This is a joint work with Pavel Tumarkin.

Friday, 15 July 2016

• 13:15, Room C01-142
  Ann Kiefer (Bielefeld): Units in Integral Group Rings via Fundamental Domains and Hyperbolic Geometry
  Abstract: The motivation of this work is the investigation on the unit group of an integral group ring U(ZG) for a finite group G. By the Wedderburn-Artin Theorem, the study of U(ZG) may be reduced, up to commensurability, to the study of SL_n(O) for n ≥ 1 and O an order in some division ring D. There exists descriptions of a finite set of generators for a subgroup of finite index in SL_n(O) for a large number of cases. Excluded from this result are the so-called exceptional components of QG.
  Our work consists in finding a presentation, for SL_n(O) associated to some of these exceptional components. In all the cases we treat, the group SL_n(O) has a discontinuous action on hyperbolic space of dimension 2 or 3, on hyperbolic space of higher dimensions, or on some product of hyperbolic spaces. By constructing fundamental domains for these discontinuous actions, we get generators for the groups in question.
• 14:30, Room C01-142
  Martin Kalck (Edinburgh): Knörrer-type equivalences for two-dimensional cyclic quotient singularities
  Abstract: We construct triangle equivalences between singularity categories of two-dimensional cyclic quotient singularities and singularity categories of a new class of finite dimensional algebras, which we call Knörrer invariant algebras. In the Gorenstein case, we recover a special case of Knörrer’s equivalence for hypersurfaces. Time permitting, we’ll explain how this led us to a formula for the Ringel duals of certain strongly quasi-hereditary algebras. This is based on joint work with Joe Karmazyn.
• 16:00, Room C01-142
  Grzegorz Bobiński (Torun): On nonsingularity in codimension one of irreducible components of module varieties over quasi-tilted algebras
  Abstract: For a given dimension vector over a triangular algebra the closure of the set of modules of projective dimension at most 1 is an irreducible component (if nonempty). There are results showing that this component should have good geometric properties. For example, if the dimension vector is the dimension vector of a directing (non necessarily indecomposable) module, then this component is nonsingular in codimension one. A new result (joint with Zwara) says that the same holds for the dimension vectors of regular modules over concealed canonical algebras. We hope to generalize these results to arbitrary dimension vectors over quasi-tilted algebras.

Friday, 01 July 2016

• 13:15, Room C01-142
  Hagen Meltzer (Szczecin): Exceptional objects for nilpotent operators with invariant subspace
  Abstract: This is a report on joint work with Piotr Dowbor (Torun) and Markus Schmidmeier (Boca Raton). We study (graded) vector spaces equipped with a nilpotent operator of nilpotency degree n and an invariant subspace. This problem is related to an old one stated by Birkhoff and recent results were obtained by Ringel-Schmidmeier, by Simson and in joint work with Kussin and Lenzing investigating stable vector bundle categories for weighted projective lines. In particular for n=6 the category is of tubular type.
  We study exceptional objects in this category and show that each of them can be exhibited by matrices having as coefficients only 0 and 1.
• 14:30, Room C01-142
  Alexander Kleshchev (Eugene): Stratifications of Khovanov-⁠Lauda-⁠Rouquier algebras
  Abstract: We review standard module theory for Khovanov-Lauda-Rouquier algebras of finite and affine types, its connections with PBW bases in quantum groups, and affine highest weight categories. Time permitting, we give an applications to blocks of symmetric groups and Hecke algebras.

Friday, 24 June 2016

• 14:15, Room C01-142
  Alexander Merkurjev (Los Angeles): Rationality problem for classifying spaces of algebraic groups
  Abstract: Many classical objects in algebra such as simple algebras, quadratic forms, algebras with involutions, Cayley algebras etc, have large automorphism groups and can be studied by means of algebraic group theory. For example, for each type of algebraic objects there is an algebraic variety (called the classifying space of the corresponding algebraic group) that classifies the objects. The simpler the structure of this variety, the simpler the classification. For example, rationality of the classifying variety means that the objects can be described by algebraically independent parameters. I will discuss the rationality property of classifying varieties.

Friday, 17 June 2016

• 13:15, Room C01-142
  George Dimitrov (Bonn): Unstable exceptional objects in hereditary categories
  Abstract: On the way of describing the entire Bridgeland stability spaces on some quivers we handled unstable exceptional objects in hereditary categories, whereby specific pairwise relations between exceptional objects were utilized. In this talk I will tell more about this.
• 14:30, Room C01-142
  Xin Fang (Cologne): On degenerations of flag varieties
  Abstract: Motivated by the PBW filtration of Lie algebras, E. Feigin defined the degenerate flag varieties, which are flat degenerations of the corresponding flag varieties. The purpose of this talk is to introduce a new family of (flat) degenerations of complete flag varieties of type A, called linear degenerate flag varieties, by classifying flat degenerations of a particular quiver Grassmannian. The geometry of these degenerations will also be presented. This talk is based on a joint work with G. Cerulli Irelli, E. Feigin, G. Fourier and M. Reineke.
• 16:00, Room C01-142
  Fritz Hörmann (Freiburg): Fibered multiderivators, (co)homological descent and Grothendieck's six operations
  Abstract: The theory of derivators enhances and simplifies the theory of triangulated categories. We propose a notion of fibered (multi-)derivator, which similarly enhances fibrations of (monoidal) triangulated categories. We present a theory of cohomological as well as homological descent in this language. The key is a generalization of the notion of ``fundamental localizer'' to diagrams in a category with Grothendieck topology. The main motivation is a descent theory for Grothendieck's six operations. We will also explain how a (classical) six functor context can be defined as a fibered multicategory, thus giving a simple precise definition including all possible compatibility relations between the six functors.

Friday, 10 June 2016

• 14:15, Room C01-142
  Moritz Groth (Bonn): Characterizations of abstract stable homotopy theories
  Abstract: The typical triangulated categories arising in nature are homotopy categories of suitable stable homotopy theories in the background. This applies to derived categories of abelian categories as well as to the stable homotopy category of spectra. In this talk we discuss various characterizations of abstract stable homotopy theories, thereby describing aspects of the calculus of chain complexes. Moreover, each of these characterizations specializes to an answer to the following question: what is the defining feature of the passage from (pointed) topological spaces to spectra?

Friday, 03 June 2016

• 14:15, Room C01-142
  William Sanders (Trondheim): A Pointless approach to triangulated categories
  Abstract: In the past several decades, algebraists have used various notions of support to study the thick subcategories of certain triangulated categories. However, each of these notions require the triangulated category in question to have additional structure, such as a Noetherian ring action or else a tensor triangulated structure. In this talk we will use pointless topology to develop a theory of supports for any triangulated category whose thick subcategories form a set. To do this, we identify a collection of thick subcategories which are in bijection with the open sets of a topological space.
  The study of a space via the lattice of open sets is called pointless topology. Since many topological spaces are completely determined by their lattice of open sets, every topological concept has a pointless, lattice theoretic analogue. Therefore, we can use pointless topology to study the lattice of thick subcategories of a triangulated category from a topological and geometric perspective.
• 15:30, Room C01-142
  Markus Schmidmeier (Boca Raton): Finite direct sums of cyclic embeddings with an application to invariant subspace varieties
  Abstract: In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. We use partial maps on Littlewood-Richardson tableaux to generalize this result to finite direct sums of such embeddings. As an application to invariant subspaces of nilpotent linear operators, we develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order. This is a report about a joint project with Justyna Kosakowska from Torun.

Friday, 20 May 2016

• 13:15, Room C01-142
  Andreas Hochenegger (Köln): Spherical subcategories
  Abstract: In a triangulated category, a spherical object is defined as a Calabi-Yau object that has a two-dimensional (graded) endomorphism ring. They are interesting as the associated twist functor gives an autoequivalence. In this talk, I will show what happens if one drops the Calabi-Yau property, illustrated by examples.
  This is joint work with Martin Kalck and David Ploog.
• 14:30, Room C01-142
  David Ploog (Berlin): Discrete triangulated categories
  Abstract: The study of discrete-derived algebras (in Vossieck's sense) exhibited some curious properties of their derived categories. E.g. dimensions of homomorphism spaces between indecomposable objects are at most 2; any two objects have only finitely many different cones; hearts of bounded t-structures have only finitely many indecomposable objects. In this talk, we look at such properties among abstract triangulated categories. (Joint work with N. Broomhead and D. Pauksztello.)
• 16:00, Room C01-142
  Ragnar-Olaf Buchweitz (Toronto): Tilting theory for one-dimensional Gorenstein algebras
  Abstract: We show that for a connected, commutative, positively graded Gorenstein algebra R of Krull dimension one wth nonnegative a-invariant there are tilting objects both for per(qgr R), the triangulated category of perfect complexes of “sheaves” on the (virtual) projective scheme underlying R, as well as for the (larger) stable category of graded maximal Cohen-Macaulay that are generically locally free.
  We’ll discuss in some detail the examples of (not necessarily reduced) line configurations in the plane, the simple curve singularities, and the curve singularities defined by symmetric numerical semigroups.
  This is based on joint work with Osamu Iyama and Kota Yamaura.
  

Friday, 13 May 2016

• 14:15, Room C01-142
  Theo Raedschelders (Brussels): Derived categories of noncommutative quadrics and Hilbert schemes of points
  Abstract: A philosophy emerging from recent work of Orlov says roughly that for a smooth projective variety X, there should be a smooth projective M_X representing a moduli problem on X such that Perf-X embeds as an admissible subcategory into Perf-M_X. Moreover, noncommutative deformations of X should embed into commutative deformations of M_X. I will discuss this philosophy and make it precise for X a smooth quadric surface and M_X the Hilbert scheme of two points on X. This is joint work with Pieter Belmans.
• 15:30, Room C01-142
  Tobias Barthel (Bonn): Algebraic approximations to stable homotopy theory
  Abstract: Viewing the stable homotopy category as a homotopical analogue of the derived category of abelian groups reveals an infinite tower of "chromatic primes" K(n,p) interpolating between characteristic 0 and characteristic p. There are many examples of phenomena in the corresponding K(n,p)-local categories that become more algebraic and homogeneous when p goes to infinity. After reviewing the required background from stable homotopy theory, I will explain joint work in progress with Schlank and Stapleton in which we construct an algebraic category that captures such generic phenomena in chromatic homotopy theory. Our methods are inspired by ideas from mathematical logic, and might be applicable in other contexts as well.

Friday, 06 May 2016

• 14:15, Room C01-142
  Rolf Farnsteiner (Kiel): Indecomposable Modules, McKay Quivers, and Ramification
  Abstract: Let $k$ be an algebraically closed field of characteristic p\ge 3. In 1991, A. Premet determined the Green ring of the restricted enveloping algebra U_0(sl(2)) and provided an explicit description of the indecomposable U_0(sl(2))-modules. Earlier work by Drozd, Fischer and Rudakov had essentially shown that the non-simple blocks of U_0(sl(2)) are Morita equivalent to the trivial extension of the path algebra of the Kronecker quiver. This implies in particular that U_0(sl(2)) is an algebra of domestic representation type. In this talk we indicate how Premet's classification can be extended to finite group schemes of domestic representation type. The combinatorial data of the stable Auslander-Reiten quiver of such group schemes are related to McKay quivers and the ramification indices associated to morphisms between certain support varieties.

Friday, 22 April 2016

• 13:15, Room C01-142
  Jan Geuenich (Bonn): Jacobian Algebras for Modulated Quivers and Triangulated Orbifolds
  Abstract: To begin with, I discuss modulations for weighted quivers in a general framework. After that, I move on to cyclic Galois modulations. I explain what form Jacobian algebras and DWZ mutation assume in this context. As an interesting application I call attention to Jacobian algebras for adjacency quivers of triangulated unpunctured orbifolds. This is joint work with Daniel Labardini Fragoso.
• 14:30, Room C01-142
  Oriol Raventós-Morera (Regensburg): Generators and descent in triangulated categories
  Abstract: The existence of a generator in a triangulated category has strong consequences. Most importantly, it is a fundamental assumption for proving representability results, which in their turn are used to show the existence of adjoint functors and duality formulas.
  In this talk, we briefly introduce different notions of generators and exhibit some new examples, especially in the case of derived categories of rings. Next we introduce the notion of decent in a triangulated category and show how it is related to the notion of generator. We explain how descent in triangulated categories can be viewed as an analogue of Grothendieck faithfully flat descent once we work with an infinity categorical enhancement of our triangulated category.
• 16:00, Room C01-142
  Peter Symonds (Manchester): Endotrivial modules for infinite groups
  Abstract: Endotrivial modules for finite groups have been extensively studied, Here we see what we can say for infinite groups. First, we have to decide on a stable category and work out for which groups it has good properties; Gorenstein projective modules appear extensively here. Then we develop some tools that can be used for calculation in some particular cases.

Friday, 15 April 2016

• 14:15, Room C01-142
  Paolo Stellari (Milano): Uniqueness of dg enhancements in geometric contexts
  Abstract: It was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper. In this talk we will explain how to extend Lunts-Orlov's results to several interesting geometric contexts. Namely, we care about the category of perfect complexes on noetherian separated schemes with enough locally free sheaves and the derived category of quasi-coherent sheaves on any scheme. These results will be compared to the existence and uniqueness of dg lifts of exact functors of geometric nature. This is a joint work with A. Canonaco.

Friday, 08 April 2016

• 14:15, Room V3-201
  Chrysostomos Psaroudakis (Trondheim): Realisation Functors in Tilting Theory
  Abstract: Let T be a triangulated category and H the heart of a t-structure in T. In this setting it is natural to ask what is the relation of T with the bounded derived category of the abelian category H. Under some assumptions on T and the t-structure, Beilinson-Bernstein-Deligne constructed a functor between these two triangulated categories, called the realisation functor. The first part of this talk is devoted to recall this construction. Then the main aim is to show how to obtain derived equivalences between abelian categories from not necessarily compact tilting and cotilting objects. The key ingredients of this result are the realisation functor and a notion of (co)tilting objects in triangulated categories that we introduce. As a particular case we explain how derived equivalences between Grothendieck categories can be realised as cotilting equivalences. This is joint work with Jorge Vitoria (arXiv:1511.02677).

Wednesday, 09 March 2016

• 10:15, Room V2-213
  Laurent Demonet (Nagoya): Algebras of partial triangulations
  Abstract: This is a report on [Dem16].
  We introduce a class of finite dimensional algebras coming from partial triangulations of marked surfaces. A partial triangulation is a subset of a triangulation.
  This class contains Jacobian algebras of triangulations of marked surfaces [LF09] (see also [DWZ08]) and Brauer graph algebras [WW85]. We generalize properties which are known or partially known for Brauer graph algebras and Jacobian algebras of marked surfaces. In particular, these algebras are symmetric when the considered surface has no boundary, they are at most tame, and we give a combinatorial generalization of flips or Kauer moves on partial triangulations which induces (in most cases) derived equivalences between the corresponding algebras. Notice that we also give an explicit formula for the dimension of the algebra.
  
  [Dem16] Laurent Demonet. Algebras of partial triangulations. arXiv: 1602.01592, 2016.
  [DWZ08] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky. Quivers with potentials and their representations. I. Mutations. Selecta Math. (N.S.), 14 (1): 59–119, 2008.
  [LF09] Daniel Labardini-Fragoso. Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. (3), 98 (3): 797–839, 2009.
  [WW85] Burkhard Wald and Josef Waschbüsch. Tame biserial algebras. J. Algebra, 95 (2): 480–500, 1985.

Wednesday, 02 March 2016

• Mini-Workshop "Triangulated Categories and Geometry"

Tuesday, 01 March 2016

• Mini-Workshop "Triangulated Categories and Geometry"

Friday, 05 February 2016

• 14:15, Room V2-213
  Mikhail Gorsky (Paris): Hall algebras with coefficients and localization of categories
  Abstract: Hall algebras provide one of the first known examples of additive categorification. They appear in the study of the representation theory of quantum groups and of counting invariants of moduli spaces. I will discuss various versions of Hall algebras of exact and triangulated categories and explain how localizations of categories can be used to construct Hall algebras with (quantum tori of) coefficients. If time permits, i will also discuss their relation to quiver varieties and quantum cluster algebras.

Thursday, 04 February 2016

• Faculty Colloquium
  17:15, Room V2-210/216
  Paul Balmer (Los Angeles): An invitation to tensor-triangular geometry
  Abstract: We will begin by an overview of the various fields where tensor-triangulated categories are commonly used, starting in topology and algebraic geometry and moving towards representation theory and beyond. Through all these areas, we shall see how the classification of objects up to the available structures leads to a geometric invariant, called the spectrum. If time permits, I shall present some new such classifications recently obtained in equivariant stable homotopy theory in joint work with Beren Sanders.

Friday, 29 January 2016

• 13:15, Room V2-213
  Rebecca Reischuk (Bielefeld): The adjoints of the Schur functor
  Abstract: The so-called Schur functor is an exact functor from the category of strict polynomial functors to the category of representations of the symmetric group. In an earlier work we have shown that this functor transfers the monoidal structure inherited from the category of divided powers to the Kronecker product on symmetric group representations. It is well-known that the Schur functor has fully faithful left and right adjoints. We show that these functors can be expressed in terms of the monoidal structure of strict polynomial functors. As an application we consider the tensor product of two simple strict polynomial functors and give a necessary and sufficient condition to be again simple.
• 14:30, Room V2-213
  Greg Stevenson (Bielefeld): Relative stable categories of finite groups
  Abstract: A few years ago Benson, Iyengar, and Krause introduced an analogue of the stable module category for representations of a finite group over any commutative ring. I will discuss some recent progress on understanding the structure of these categories (coming from joint work with Baland, and Baland and Chirvasitu).
• 16:00, Room V2-213
  Jon Carlson (Athens, Georgia): Group algebras and Hopf algebras
  Abstract: This lecture concerns an effort to resolve a technical issue that arises in attempt to make connections between the areas of group representation theory and commutative algebra. The difficulty is that while there are functors between the module categories that are very useful, the coalgebra stuctures do not match up. This will be demonstrated in explicit detail. Even though the difficulty has been treated many time in the literature, it has a rather easy partial solution that was missed previously. This is joint work with Srikanth Iyengar.

Friday, 22 January 2016

• 14:15, Room V2-213
  Matthew Pressland (Bath): Internally Calabi-Yau algebras and cluster-tilting objects
  Abstract: Cluster categories, which are 2-Calabi–Yau triangulated categories containing cluster-tilting objects, have played a significant role in understanding the combinatorics of cluster algebras without frozen variables. When frozen variables do occur, an analogous categorical model may be provided by a Frobenius category whose stable category is 2-Calabi-Yau, although such a categorification is only known in a few cases. It is observed by Keller-Reiten that the endomorphism algebra of a cluster-tilting object in such a category has a certain relative, or internal, Calabi-Yau symmetry. In this talk, I will explain how to go in the opposite direction; given an algebra A with a suitable level of Calabi-Yau symmetry, I will explain how to construct a Frobenius category admitting a cluster-tilting object with endomorphism algebra A.
• 15:30, Room V2-213
  Igor Burban (Cologne): Singular curves and quasi-⁠hereditary algebras
  Abstract: In my talk (based on a joint work with Yu. Drozd and V. Gavran), I shall describe a certain non-commutative resolution of singularities of a reduced algebraic curve X.
  Nice homological properties of this resolution imply several new results on the Rouquier dimension of the derived category of coherent sheaves on X. Moreover, in the case X is rational and projective, this construction allows to construct a finite dimensional quasi–hereditary algebra A such that the triangulated category Perf(X) embeds into D^b(A-mod) as a full subcategory.
  

Friday, 15 January 2016

• 14:15, Room V2-213
  Sebastian Klein (Antwerpen): Relative tensor triangular Chow groups and applications
  Abstract: In my previous talk in the BIREP seminar, I introduced a notion of Chow groups for tensor triangulated categories. This time, after a brief reminder, I will introduce a generalization of the concept which allows us to consider different types of triangulated categories: we can look at 'big' triangulated categories which do not necessarily admit a monoidal structure themselves but only an action by a tensor triangulated category. As applications, we recover the Chow groups of a possibly singular algebraic variety from its homotopy category of quasi-coherent injective sheaves, we construct localization sequences associated to the restriction to an open subset and we are able to define triangular Chow groups of 'noncommutative ringed schemes'.

Friday, 18 December 2015

• 14:15, Room V2-213
  Philipp Lampe (Bielefeld): On singular loci for cluster algebras of type D
  Abstract: Muller, Rajchgot and Zykoski have computed the singular locus of a cluster algebra of type A. We complement their work and compute the singular locus of a cluster algebra of type D. Especially, we describe the defining ideal of the singular locus by non-prime cluster variables.
• 15:30, Room V2-213
  Henning Krause (Bielefeld): The variety of subadditive functions for finite group schemes
  Abstract: For a finite group scheme G, Friedlander and Pevtsova introduced pi-points which give rise to certain endofinite 'point modules'. Using then Crawley-Boevey's correspondence between endofinite modules and subadditive functions on finitely presented modules, it is possible to recover the projective variety of the cohomology of G from the equivalence classes of subadditive functions. This talk is based on joint work with Benson, Iyengar and Pevtsova.

Friday, 11 December 2015

• 14:15, Room V2-213
  Shraddha Srivastava (Chennai): Strict polynomial functors and the Kronecker product
  Abstract: Strict polynomial functors of degree d provide a unified way of studying polynomial representations of degree d of the group schemes GL(n), for all n. A priori the category of polynomial representations of GL(n) of degree d has no internal tensor product, as well as no internal hom. H. Krause discovered an internal tensor for strict polynomial functors via Day convolution. Though the descriptions of the same internal hom for strict polynomial functors by A. Touze and by H. Krause differ, both are useful. The internal hom and internal tensor were used to establish Ringel duality and Koszul duality for strict polynomial functors respectively. Both the authors also gave several examples of the internal tensor product/hom and raised the question of computing it explicitly. There is a wellknown functor, namely the Schur functor, from strict polynomial functors to the symmetric group representations. I will show that the Schur functor preserves the tensor product on each side. I will also show some explicit computations of the internal tensor product involving divided powers, symmetric powers, exterior powers and Weyl functors. An example of calculating the Kronecker multiplicities via this treatment will be discussed. This is joint work with Upendra Kulkarni and K.V. Subrahmanyam.
• 15:30, Room V2-213
  Alexandra Zvonareva (St. Petersburg): On the computation of derived Picard groups
  Abstract: The derived Picard group of an algebra is the group of isomorphism classes of two-sided tilting complexes, or equivalently the group of standard autoequivalences of the derived category modulo natural isomorphisms. In this talk I will discuss how silting mutations, orbit categories and spherical objects can be used to obtain a description of the derived Picard group on the example of a selfinjective Nakayama algebra. This talk is based on joint work with Yury Volkov.

Saturday, 05 December 2015

• Workshop on Representations, Support, and Cohomology

Friday, 04 December 2015

• Workshop on Representations, Support, and Cohomology

Friday, 27 November 2015

• 14:15, Room V2-213
  Hans Franzen (Bonn): Donaldson-⁠Thomas invariants of quivers via Chow groups of quiver moduli
  Abstract: We use a presentation of Chow rings of (semi-)stable quiver moduli to show that the primitive part of the Cohomological Hall algebra of a symmetric quiver is given by Chow groups of moduli spaces of simple representations. This implies that the DT invariants agree with the dimensions of these Chow groups.
  

Friday, 20 November 2015

• 13:15, Room V2-213
  Alfredo Najera Chavez (Bonn): Frobenius orbit categories and categorification of cluster algebras
  Abstract: In this talk I will present some general results on orbit categories associated to Frobenius categories. We will apply these result to the context of Nakajima categories associated to Dynkin quivers to obtain a categorification of families of finite type skew-symmetric cluster algebras with coefficients. As a consequence we obtain a description of the category of Cohen-Macaulay modules over certain isolated singularities as the completed orbit category of a Nakajima category.
• 14:30, Room V2-213
  Magnus Engenhorst (Bonn): Maximal green sequences for quiver categories
  Abstract: Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. A third incarnation are maximal chains in the Hasse quiver of torsions classes. More generally, we introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. In the case of preprojective algebras we show that a quiver has a maximal green sequence if and only if it is of Dynkin type.
• 16:00, Room V2-213
  Raymundo Bautista (Morelia): Differential tensor algebras, boxes, and exact structures
  Abstract: A box B consists of an algebra A over some field, an A-A bimodule U with a co-associative co-multiplication, and a co-unit. The theory of representations of boxes has been an important tool in the representation theory of finite dimensional algebras over algebraically closed fields. We can mention the tame-wild dichotomy proved by Y. Drozd and the discovery due to W. Crawley-Boevey of generic modules and its important role in the tame representation type. Given a box as before, one can define a differential on the tensor algebra of A over the A-A bimodule given by the kernel of the co-unit (an A-A bimodule morphism from U to A), this give us a graded differential algebra. Then one can define a category of representations of this algebra and a class of pairs of composable morphisms. In some cases this class is an exact structure. We will see the connection (discovered by S. Koenig, J. Kulshammer and S. Ovsienko ) with quasi-hereditary algebras and the category of modules with standard filtration.

Friday, 13 November 2015

• 14:15, Room V2-213
  Thomas Gobet (Kaiserslautern): On twisted filtrations on Soergel bimodules
  Abstract: The Iwahori-Hecke algebra of a Coxeter group has a standard and a costandard basis, as well as two canonical bases. If the Coxeter group is finite, it was shown by Dyer that the product of an element of the canonical basis with an element of the standard basis has positive coefficients when expressed in the standard basis. Using Dyer’s notion of biclosed sets of reflections, we consider a family of bases containing both the standard and costandard bases and show that an element of the canonical basis has a positive expansion in any basis from this family. The key tool for this is to consider twisted filtrations on Soergel bimodules (these bimodules categorify the canonical basis of the Hecke algebra) and interpret the coefficients as multiplicities in these filtrations. This generalizes Dyer’s result to a more general family of bases as well as to arbitrary Coxeter groups. Elements of these bases turn out to be images of Mikado braids as introduced in a joint work with F. Digne. It time allows, we will mention a conjecture on the Rouquier complexes of these braids, which would imply a generalized inverse Kazhdan-Lusztig positivity.

Friday, 06 November 2015

• 14:15, Room V2-213
  Magdalena Boos (Wuppertal): Criteria for finite parabolic conjugation
  Abstract: Motivated by the study of commuting varieties, we consider a parabolic subgroup P of GLn and study its conjugation-action on the variety of nilpotent matrices in LieP. The main question posed in this talk is "For which P does the mentioned action only admit a finite number of orbits?" In order to approach a finiteness criterion which answers our main question, we look at covering quivers, quadratic forms, Delta-filtrations and more. (This is work in progress, joint with M. Bulois)
• 15:30, Room V2-213
  Paul Balmer (Los Angeles): Endotrivial representations of finite groups and equivariant line bundles on the Brown complex
  Abstract: I will explain what endotrivial representations are and how they relate to the equivariant line bundles on the Brown complex of non-trivial p-subgroups. Some time will be spent introducing the Brown complex and related basic questions.

Friday, 30 October 2015

• 14:15, Room V2-213
  Dirk Kussin (Paderborn): Infinite-dimensional modules over tubular algebras
  Abstract: We report on joint work with Lidia Angeleri. For a (concealed canonical) tubular algebra we will focus on modules of a given real slope, in particular on (large) tilting or cotilting modules, and on pure-injective modules.
• 15:30, Room V2-213
  Helmut Lenzing (Paderborn): An interesting class of hereditary categories
  Abstract: Let X be a weighted projective line of tubular weight type (2,3,6), (2,4,4), (3,3,3) or (2,2,2,2). Let H be the category of coherent sheaves on X. For each irrational real number r, we form the full subcategory H<r> of the bounded derived category D^b(H) of coherent sheaves on X, assembling all indecomposables of slope < r from H and all indecomposables of slope > r from H[-1]. This yields a category H<r> that is Hom-finite abelian hereditary with Serre duality, where the Serre functor is an equivalence; moreover each tubular algebra B of the same weight type is realizable by a tilting object in H<r>. Moreover, two such categories H<r> and H<s> are equivalent if and only if s=(ar+b)/(cr+d) for integers a, b, c, d satisfying ad-bc=1, thus resulting in uncountably many nonequivalent categories of type H<r>. Conjecturally, the category H<r> plays a key role in investigating the category of indecomposable quasi-coherent sheaves (resp. indecomposable infinite dimensional B-modules) of irrational slope r, a problem attacked by Harland and Prest during the last years through model-theoretic methods.

Friday, 23 October 2015

• 14:15, Room V2-213
  Baolin Xiong (Beijing): Generalized monomorphism categories
  Abstract: In this talk, we will introduce the generalized monomorphism category, which is a generalization of the submodule category of Ringel-Schmidmeier and the monomorphism category of X.W.Chen, P.Zhang and his coauthors. We will view the submodule category and the monomorphism category again from the point of homological algebra. Some basic properties of the generalized monomorphism category will be given. This is a joint work with W.Hu and X.H.Luo.
• 15:30, Room V2-213
  Julia Sauter (Bielefeld): On quiver Grassmannians and orbit closures for representation-finite algebras
  Abstract: We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra.
  For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This is joint work with William Crawley-Boevey and it generalizes results of Cerulli Irelli, Feigin and Reineke.

Friday, 17 July 2015

• 14:15, Lecture Hall H11
  Alexander Nenashev (Toronto): Homological algebra in p-exact categories
  Abstract: This is a survey talk on homological algebra in a non-additive setting.

Friday, 03 July 2015

• 13:15, Lecture Hall H2
  Jesse Burke (Los Angeles): Generalized Koszul duality
  Abstract: We will discuss a generalization of Koszul duality, as formulated by Keller, Lefevre-Hasegawa and Positselski, to the case of A-infinity algebras defined over a commutative ring, with special attention to the definition of the categories involved. We will also discuss conjectural applications to representation theory.
• 14:30, Lecture Hall H2
  Nathan Broomhead (Hannover): Thick subcategories of discrete derived categories
  Abstract: I will describe some work in progress, attempting to understand the lattice of thick subcategories of discrete derived categories and of derived categories of extended A type, using collections of exceptional and sphere-like objects.

Friday, 12 June 2015

• 14:00, Room U2-113
  Philipp Lampe (Bielefeld): Almost periodic sequences attached to non-crystallographic root systems
  Abstract: We study Fomin-Zelevinsky’s mutation rule in the context of non-crystallographic root systems. In particular, we construct almost periodic sequences of real numbers for the non crystallographic root systems of rank 2 by adjusting the exchange relation for cluster algebras. Moreover, we describe a matrix mutation class in rank 3.
• 15:15, Room U2-113
  Lutz Hille (Münster): Derived equivalences between GL-weighted projective spaces and resolutions of quotients of toric almost Fano varieties
  Abstract: The endomorphism algebra of a tilting bundle on a projective variety of dimension d is between d and 2d by a recent result with Buchweitz. The case of global dimension d is of particular interest. We give an infinite list of varieties allowing a tilting bundle consisting of line bundles with this property using reflexive simplices and toric varieties. Given a reflexive simplex \Delta we construct a stack that is a GL weighted projective space X(\Delta) together with a tilting bundle consisting of line bundles and an action of a finite abelian group G. We also consider a resolution Y of the corresponding singular toric variety associated to \Delta. Then we claim that X(\Delta) and Y have isomorphic derived categories as follows. On both varieties we have a tilting bundle consisting of line bundles with the same endomorphism algebra A of global dimension d.
  
  In the talk we start with several motivations for such a result: Bridgelands flops, a conjecture of Bondal and Orlov, the recent joint work with Buchweitz, to mention some of them. Then we explain the main construction using polytopes. Finally we discuss several applications.

Monday, 08 June 2015

• 14:15, Lecture Hall H6
  Mikhail Kapranov (New Haven): Perverse Schobers and Fukaya categories with coefficients for punctured surfaces
  Abstract: The (instanton-⁠less approximation to the) Fukaya category of a punctured Riemann surface can be seen as a categorification of the relative first homology group with constant coefficients. The talk, based on joint works and projects with T. Dyckerhoff, V. Schechtman and Y. Soibelman, will explain this construction as well as an extension to the case of variable coefficients. The role of coefficient data is provided by perverse Schobers, which are conjectural categorical analogs of perverse sheaves, analogs which can be made precise sense for the case of surfaces.

Wednesday, 27 May 2015

• 10:15, Room V3-201
  Travis Mandel (Aarhus): Tropical curve counting and canonical bases
  Abstract: Gross, Hacking, Keel, and Kontsevich recently constructed certain canonical bases for cluster algebras. The construction is combinatoric, but the bases are conjecturally controlled by the Gromov-Witten theory of the mirror cluster variety. I will discuss a new construction of these bases in terms of certain tropical curve counts which one expects to correspond to the predicted holomorphic curve counts. I will also discuss a refinement of the tropical counts which produces quantized versions of the canonical bases.

Friday, 22 May 2015

• 14:00, Room E0-180
  Sira Gratz (Hannover): Cluster algebras of infinite rank as colimits
  Abstract: We formalize cluster algebras of infinite rank by showing that, in the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler, every rooted cluster algebra can be written as a directed colimit of rooted cluster algebras of finite rank.
• 15:15, Room E0-180
  Sarah Scherotzke (Bonn): Quiver varieties and self-injective algebras
  Abstract: We introduce a new class of quiver varieties, recovering as special cases the cyclic and classical Nakajima quiver varieties. We show that the geometry of the new quiver varieties is closely linked to the representation theory of a suitable finitely-generated algebra P, which is self-injective if Q is of ADE Dynkin type. The affine quiver variety, defined as GIT quotient, is shown to be isomorphic to the moduli space of representations of a finitely-generated algebra S. The algebra S specializes to the preprojective algebra if we consider classical quiver varieties. Finally, we use our results to construct desingularisations of quiver Grassmannians of modules of self-injective algebras of finite type.

Friday, 15 May 2015

• 13:15, Room T2-233
  Raf Bocklandt (Amsterdam): Gentle A_infinity Algebras and Mirror Symmetry
  Abstract: We introduce the notion of gentle A-infinity algebras and discuss their representation theory in terms of strings and bands. We relate this to mirror symmetry for Riemann surfaces and explain the connection with SYZ-fibrations and Jenkin-Strebel differentials.
• 14:30, Room T2-233
  Julia Pevtsova (Seattle): Varieties of elementary subalgebras of modular Lie algebras
  Abstract: Motivated by questions in representation theory, Carlson, Friedlander and the speaker instigated the study of projective varieties of abelian p-nilpotent subalgebras of a fixed dimension r for a p-Lie algebra g. These varieties are close relatives of the much studied class of varieties of r-tuples of commuting p-nilpotent matrices which remain highly mysterious when r>2. In this talk, I¹ll present some of the representation-theoretic motivation behind the study of these varieties and describe their geometry in a very special case when it is well understood: namely, when r is the maximal dimension of an abelian p-nilpotent subalgebra of g for g a Lie algebra of a reductive algebraic group. This is joint work with J. Stark.
• 16:00, Room T2-233
  Jon Carlson (Athens, Georgia): Thick subcategories of the relative stable category
  Abstract: Let G be a finite group and k an algebraically closed field of characteristic p > 0. Let H be a collecction of p-subgroups of G. We investigate the relative stable category stmod_H(kG) of finitely generated modules modulo H-projective modules. Triangles in this category correspond to H-split sequences. Hence, compared to the ordinary stable category there are fewer triangles and more thick subcategories. In this talk we describe several methods to construct thick tensor ideal subcategories. This is work in progress.

Wednesday, 13 May 2015

• 10:15, Room V5-227
  Rasool Hafezi (Isfahan): On relative derived category
  Abstract: In this talk, I will introduce relative derived category and then discuss about its properties and its connection with ordinary derived category. If time permits, I will explain a triangle equivalence between a sub-triangulated category of homotopy category of Gorenstein projective modules and a localization of homotopy category of acyclic complex of projective modules.

Tuesday, 17 March 2015

• 14:15, Room V2-200
  Steven Sam (Berkeley): Some examples of representation stability
  Abstract: I'll explain some recent joint work with Andrew Snowden and Andrew Putman that give proofs for some results in combinatorics, topology, and algebra by interpreting them as finite generation statements for algebraic structures. Some examples are the Lannes-Schwartz artinian conjecture and Stembridge's conjecture on stability of Kronecker coefficients of the symmetric group.

Friday, 06 February 2015

• 14:00, Lecture Hall X-E0-202
  Hanno Becker (Bonn): Models for singularity categories and applications to knot invariants
  Abstract: This talk is a summary of the results of my PhD project. First, I will outline the construction of model categorical enhancements of singularity categories within the framework of abelian model structures and cotorsion pairs. Afterwards, I will explain how a suitable model structure on the category of linear factorizations (enhancing the homotopy category of matrix factorizations) can be used to obtain a description of the Khovanov-Rozansky knot invariant in terms of Hochschild homology of Soergel bimodules.
• 15:15, Lecture Hall X-E0-202
  Martin Brandenburg (Münster): Tensor categorical foundations of algebraic geometry
  Abstract: I will talk about tensor categorical algebraic geometry, a theory which internalizes (universal) constructions from commutative algebra and algebraic geometry into cocomplete tensor categories. This is motivated by the fact that most schemes (in fact, algebraic stacks) may be reconstructed from their cocomplete tensor category of quasi-coherent sheaves. The theory resembles noncommutative algebraic geometry (but with additional tensor products) and topos theory (where the cartesian product is replaced by a tensor product). Specifically I will talk about tensor-categorical analogs of affine schemes, projective schemes, tangent bundles and fiber products.

Friday, 30 January 2015

• 14:00, Lecture Hall X-E0-202
  Rebecca Reischuk (Bielefeld): Translating the tensor product of symmetric group representations to those of Schur algebras
  Abstract: The categories of strict polynomial functors, representations of Schur algebras and symmetric groups are strongly related to one another. In fact, for well-chosen parameters these categories are equivalent. All three categories have been intensely investigated and many properties have been translated between them. Recently another such comparison was made, namely between an internal tensor product for strict polynomial functors and the tensor product (sometimes called "Kronecker product") of symmetric group representations.
  
  In this talk we will give an overview of these categories and expose how certain representations and structures are related to one another. In particular, we will translate the "Kronecker product" to the representations of Schur algebras and explain how the quasi-hereditary structure might be a useful tool to understand the internal tensor product in more detail.
• 15:15, Lecture Hall X-E0-202
  Simone Virili (Padova): Length functions on Grothendieck categories with applications to infinite group representations
  Abstract: Let C be a Grothendieck category. A non-negative real function L (which may attain infinity) on the objects of C is a length function if L(0)=0, L is additive on short exact sequences and it is continuous on direct unions of sub-objects.
  
  In the first part of the talk we will describe a complete classification of the length functions on a Grothendieck category C with Gabriel dimension. In fact, we will see that any length function can be constructed as a linear combination of a family of "atomic functions" that arise from the composition length in some particular localization of C at a given point in the Gabriel spectrum.
  
  In the second part of the talk we will concentrate on categories of modules. More specifically, we will give partial answers to the following question: given a ring R, a length function L on Mod(R), a (infinite) group G and a crossed product group algebra R*G, when is it possible to "extend" L to a length function L' on Mod(R*G) such that L'(M\otimes_R R*G)=L(M), for all M in Mod(R)? Roughly speaking, we will see that "many" length functions of Mod(R) can be extended to "large" subcategories of Mod(R*G), provided the group G is amenable.
  
  In the last part of the talk we will illustrate some applications to classical conjectures in the representation theory of infinite groups.

Friday, 16 January 2015

• 14:00, Lecture Hall X-E0-202
  Nathan Broomhead (Hannover): Discrete derived categories and Bridgeland stability
  Abstract: Discrete derived categories, as defined by Vossieck, form a class of triangulated categories which are sufficiently simple to make explicit computation possible, but also non-trivial enough to manifest interesting behaviour. In this talk, I will explain what they are, and talk about some recent work with D. Pauksztello and D. Ploog, in which we use a CW complex constructed from silting objects to understand the corresponding spaces of Bridgeland stability conditions.
• 15:15, Lecture Hall X-E0-202
  Stefan Schröer (Düsseldorf): Brauer groups for quiver moduli
  Abstract: For several moduli spaces of stable quiver representations, we compute the Brauer group and determine the obstruction to the existence of universal quiver representations (joint with Markus Reineke).
  

Friday, 09 January 2015

• 14:00, Lecture Hall X-E0-202
  Wassilij Gnedin (Cologne): Harish-Chandra modules of SL(2,R)
  Abstract: At the ICM 1970 Gelfand posed the problem to classify the indecomposable finite-dimensional representations of a certain quiver with relations and oriented cycles. Gelfands problem was motivated by the study of Harish-Chandra modules of the Lie group SL(2,R), and attracted a lot of interest in the sequel.
  
  My talk is concerned with the explicit combinatorics of the indecomposable representations of the Gelfand quiver, their projective resolutions, their basic invariants (like Jordan-Hölder-multiplicities, top and socle) as well as their Auslander-Reiten translations in the derived category and their contragredient duals.
  
  This talk is based on joint work with Igor Burban.
• 15:15, Lecture Hall X-E0-202
  Jörg Schürmann (Münster): Twisted invariants of symmetric products of complex algebraic varieties
  Abstract: We give a new proof of formulae for the generating series of (Hodge) invariants of symmetric products X^(n) with coefficients, which hold for complex quasi-projective varieties X with any kind of singularities, and which include many of the classical results in the literature as special cases. Our proof applies to more general situations and is based on equivariant Künneth formulae and pre-lambda structures on the coefficient theory of a point, which is the Grothendieck group of a Karoubian Q-linear tensor category.
  This is joint work with Laurentiu Maxim.

Friday, 05 December 2014

• 14:00, Lecture Hall X-E0-202
  Zhi-Wei Li (Bielefeld): Completeness of cotorsion pairs in exact categories
  Abstract: We discuss a generalized version of Quillen's small object argument in arbitrary categories. We use it to give a criterion of the completeness of cotorsion pairs in arbitrary exact categories, which is a generalization of a recent result due to Saorin and Stovicek. This criterion allows us to recover Gillespie's recent work on the relative derived categories of Grothendieck categories.
• 15:15, Lecture Hall X-E0-202
  Lutz Hille (Münster): Tilting modules for Dynkin quivers of type A, Catalan numbers, and root polytopes
  Abstract: We recall the classification of tilting modules for a path algebra over a quiver of A. The number of tilting modules is just the Catalan number. We generalize the classification to cluster tilting modules and 2-support tilting modules and give an interpretation in terms of the volume of certain polytopes. These polytopes come in two series, one is defined just using the root system, the other series uses the construction of the fan of the tilting modules. The main theorem claims, that both series coincide. We conclude with a generalization of this construction using strong exceptional sequences, instead of tilting modules. Then the classification is much more elementary, however we have to consider a certain extension of the root system for the first series.

Saturday, 15 November 2014

• Maurice Auslander Memorial Workshop

Friday, 14 November 2014

• Maurice Auslander Memorial Workshop

Thursday, 13 November 2014

• Maurice Auslander Memorial Workshop

Friday, 07 November 2014

• 13:15, Lecture Hall X-E0-202
  Alexander Soibelman (Bonn): Quiver representations, parabolic connections, and the Deligne-Simpson problem
  Abstract: The additive and multiplicative formulations of the Deligne-Simpson problem ask, respectively, if a collection of complex matrices with prescribed conjugacy classes has sum 0 or product the identity. Both versions may be restated as a single question about the existence of a regular singular connection on the projective line. We approach this question by generalizing Crawley-Boevey's moment map construction for quivers to representations of squid algebras and by using a technical property coming from Beilinson and Drinfeld's work on the geometric Langlands correspondence.
• 14:30, Lecture Hall X-E0-202 (45 minutes)
  Dirk Kussin (Paderborn): Noncommutative smooth projective curves (Part I)
  Abstract: The dimension of the function (skew-) field of a noncommutative curve over its centre is a (global) measure for its noncommutativity; the square root of this number we call the skewness of the curve. We present a skewness equation, which states, that (over a perfect base field) for each point of the curve the skewness is a product of three certain numbers, each of which is a kind of local measure of noncommutativity in this point. One of these three numbers, the tau-multiplicity, is expressed in terms of the Auslander-Reiten translation. We explain links to the theory of maximal orders and ramifications. We will give examples over the real numbers. We show that the noncommutative real smooth projective curves, if not commutative, are given by the (which we call) Witt surfaces, going back to work of Ernst Witt in 1934. In this context points of non-trivial tau-multiplicity occur naturally. As a special case we study the noncommutative real elliptic curves and present a Witt surface which is a noncommutative Fourier-Mukai partner of the Klein bottle.
• 15:45, Lecture Hall X-E0-202 (45 minutes)
  Dirk Kussin (Paderborn): Noncommutative smooth projective curves (Part II)
  Abstract: See part I.

Friday, 31 October 2014

• 13:15, Lecture Hall X-E0-202
  Ryo Kanda (Nagoya): Atom-molecule correspondence for Grothendieck categories
  Abstract: For a right noetherian ring, there exists a canonical surjective map from the set of isomorphism classes of indecomposable injective modules to the set of two-sided prime ideals. Moreover, Gabriel showed that this surjection has a canonical splitting. In this talk, we will construct these maps for a class of Grothendieck categories. The maps are realized as maps between two kinds of spectra of a Grothendieck category, the atom spectrum and the molecule spectrum. In this attempt, we will find some properties of a right noetherian ring.
• 14:30, Lecture Hall X-E0-202
  Gustavo Jasso (Bonn): n-abelian categories
  Abstract: In this talk I will introduce n-abelian categories and discuss their basic properties. These categories, which are higher analogs of abelian categories, arise naturally in the study of higher Auslander-Reiten theory. If time permits, I will describe a connection with Geiss-Keller-Oppermann's (n+2)-angulated categories.
• 16:00, Lecture Hall X-E0-202
  Yu Zhou (Bielefeld): Spherical twists and intersection formulae from decorated marked surfaces
  Abstract: For each triangulation T of a decorated marked surface S, there is an associated differential graded algebra whose finite dimensional derived category D is a triangulated 3-Calabi-Yau category. In this talk, we give a correspondence between spherical twists on spherical objects in D and braid twists on closed curves in S and show that it is independent of the choice of the triangulation T. We prove two equalities between intersection numbers of curves in S and dimensions of graded morphism spaces in D. This is a joint work in progress with Yu Qiu.

Friday, 17 October 2014

• 14:00, Lecture Hall X-E0-202
  Chao Zhang (Bielefeld): Brauer-Thrall type theorems for derived category
  Abstract: This is a joint work with Yang Han. In this talk, I will introduce the numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras). Then I will define derived bounded algebras and strongly derived unbounded algebras with cohomological range. The first and second Brauer-Thrall type theorems for the bounded derived category of a finite-dimensional algebra over an algebraically closed field are obtained. The first Brauer-Thrall type theorem says that derived bounded algebras are just derived finite algebras. The second Brauer-Thrall type theorem says that an algebra is either derived discrete or strongly derived unbounded, but not both. Moreover, piecewise hereditary algebras and derived discrete algebras are characterized as the algebras of finite global cohomological width and finite global cohomological length respectively.
  
• 15:15, Lecture Hall X-E0-202
  Estanislao Herscovich (Buenos Aires): Hochschild (co)homology and Koszul duality
  Abstract: In this talk we will discuss a particular relationship between Hochschild (co)homology and the theory of Koszul duality. More particularly, we will essentially show that the Tamarkin-Tsygan calculus of an Adams connected augmented dg algebra and of its Koszul dual are dual. This uses the fact that Hochschild cohomology and homology may be regarded from a twisted construction of some natural (augmented) dg algebras and dg modules over the former. In particular, from these constructions it follows that the computation of the cup product on Hochschild cohomology and cap product on Hochschild homology of a Koszul algebra is directly computed from the coalgebra structure of the Tor(k,k) group (the first of these results is proved differently by R.-O. Buchweitz, E. Green, N. Snashall and O. Solberg). At the end of the talk, if time allows, we shall at least state how we even generalize this situation by studying twisting theory of A_infinity-algebras to compute the algebra structure of Hochschild cohomology of more general algebras. Further details and references about this can be found in the prepublication of the arXiv http://arxiv.org/abs/1405.2247.

Wednesday, 13 August 2014

• 14:15, Room V2-210/216
  Gaohong Wang (London, Ontario): Ghost numbers of group algebras
  Abstract: The generating hypothesis can be generalized to a triangulated category and has been studied for the stable module category of a group algebra of a finite group. Since the generating hypothesis fails in the stably module category in most cases, we study the ghost number of the group algebra to test its failure. This also provides an new invariant of the group algebra. We will start with some background on the stable module category and then get to some results on ghost numbers.

Friday, 18 July 2014

• 14:00, Lecture Hall H8
  Ilke Canakci (Leicester): From labelled snake graphs to abstract snake graphs, skein relations in terms of snake graphs, and their applications
  Abstract: This talk will focus on abstract snake graphs and certain relations among them which were inspired by labelled snake graphs associated to surface triangulations. Labelled snake graphs are constructed from the crossing pattern of arcs in triangulated surfaces whereas the notion of abstract snake graphs is untied from the geometry of surfaces and is given in an elementary way. Furthermore, the skein relations, which give a formula for the product of cluster variables, can be interpreted in this abstract setting. Finally, this talk will cast two applications where the snake graph calculus is the main ingredient: One of them shows the cluster algebra and the upper cluster algebra coincide for certain surfaces and the other one gives a dimension formula for the extension space of string modules associated to Jacobian algebras arising from (unpunctured) surface triangulations.
  
• 15:15, Lecture Hall H8
  Michael Gekhtman (Notre Dame): Cluster Structures on Poisson-Lie Groups
  Abstract: Coexistence of diverse mathematical structures supported on the same variety often leads to deeper understanding of its features. If the manifold is a Lie group, endowing it with a Poisson structure that respects group multiplication (Poisson– Lie structure) is instrumental in a study of classical and quantum mechanical systems with symmetries. On the other hand, the ring of regular functions on certain Poisson varieties can have a structure of a cluster algebra. I will discuss results and conjectures on natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson–Lie structures compatible with these cluster structures. Much of this talk is based on an ongoing collaboration with M. Shapiro and A. Vainshtein.

Friday, 11 July 2014

• 13:15, Lecture Hall H8
  Moritz Groth (Nijmegen): Abstract representation theory of Dynkin quiver of type A
  Abstract: Important functors in the study of representation theory of Dynkin quivers of type A are given by reflection functors, Coxeter functors, and Serre functors. In the context of representations over a field it is well-known that these functors can be realized as certain derived tensor product or hom functors.
  In this project (which is joint with Jan Stovicek) we extend these results to the context of an arbitrary abstract stable homotopy theory, including the differential-graded and the spectral context. More specifically, we establish a general theorem which guarantees that a large class of morphisms of stable derivators can be realized by spectral bimodules. An application to the context of Dykin quivers of type A yields spectral refinements of the classical chain complexes over a field. We also construct coherent Auslander-Reiten quivers, allowing us to identify Coxeter functors as some kind of spectral Nakayama functors.
• 14:30, Lecture Hall H8
  Xiao-Wu Chen (Hefei): An introduction to Beck's theorem
  Abstract: We will report on Beck's theorem, which gives a characterization of the module category of a monad. It has two applications on the equivariantization of an abelian category with respect to a finite group action. These are applied to give a complete and uniform proof of a result of Lenzing: the category of coherent sheaves on a weighted projective line of tubular type is equivalent to the category of equivariant coherent sheaves on an elliptic curve with respect to some finite abelian group action. This is joint with J. Chen and Z. Zhou in Xiamen University.

Friday, 04 July 2014

• 14:00, Lecture Hall H8
  Grzegorz Bobiński (Torun): On singularities for orbit closures for Dynkin quivers of type D
  Abstract: When studying orbit closures of representations of quivers, it is a nontrivial task to describe tangent spaces and, in particular, to determine if a given point is nonsingular. The problem lies in the fact that, in general, there is no representation theoretic interpretation of equations describing orbit closures. On the other hand, there exist natural schemes, which are defined in terms of hom-spaces and whose reduced structures coincide, in the case of Dynkin quivers, with those of orbit closures. Moreover, Riedtmann and Zwara have proved that these schemes are reduced if a quiver is of type A. In my talk I will present a joint work in progress with Zwara on type D.
• 15:15, Lecture Hall H8
  Alex Martsinkovsky (Boston): Direct summands of homological functors on length categories
  Abstract: In his La Jolla paper on coherent functors, M. Auslander described injective objects in certain functor categories as direct summands of the covariant functors Ext^1(A, —), and conjectured that they are all of that same form. He established that result in the case A was of finite projective dimension. In the same volume, P. Freyd gave a positive answer in the case the underlying abelian category has denumerable sums. Later, Auslander gave a unifying proof of these results, but also showed that the conjecture is not true in general. In this talk, we give a positive answer to the conjecture in the seemingly overlooked case when A is an object of finite length. In fact, our result is established for any additive bifunctor whose endomorphisms lift to endomorphisms of the fixed argument. That this is the case for the Ext-functor is a consequence of the Hilton-Rees theorem, for which we give a short proof. Other immediate applications include Hom modulo projectives, and, when the fixed arguments are restricted to finitely presented modules, the functors Tor_1(A,—).

Saturday, 14 June 2014

• Workshop "Non-crossing partitions in representation theory"

Friday, 13 June 2014

• Workshop "Non-crossing partitions in representation theory"

Thursday, 12 June 2014

• Workshop "Non-crossing partitions in representation theory"

Friday, 06 June 2014

• 13:15, Lecture Hall H8
  Ivo Dell'Ambrogio (Lille): Grothendieck-Neeman duality and Wirthmüller isomorphisms
  Abstract: In this talk I will review Neeman's proof of Grothendieck duality in algebraic geometry and Fausk-Hu-May's approach to the Wirthmüller isomorphism in equivariant stable homotopy, and expose their intimate relationship. More generally, we are led to study the existence and properties of adjoints to a given coproduct-preserving exact tensor functor between two rigidly-compactly generated tensor triangulated categories. A surprising outcome of this analysis is the following trichotomy result: we can only have three consecutive adjoints, or five, or infinitely many in both directions. This is joint work with Paul Balmer and Beren Sanders.
• 14:30, Lecture Hall H8
  Alex Martsinkovsky (Boston): Asymptotic stabilization of the tensor product
  Abstract: This is joint work with my student Jeremy Russell, providing an analog of Buchweitz's construction of Tate cohomology for the tensor product.
• 16:00, Lecture Hall H8
  Gus Lehrer (Sydney): R-matrices, cellularity, and tilting modules at roots of unity
  Abstract: R-matrices permit the construction of homomorphisms from the group ring of the braid group to the endomorphism algebras of tensor representations of quantum groups. This is exploited for the higher Weyl modules of quantum sl_2 to describe the relevant endomorphism algebras as cellular subalgebras of a large Temperley-Lieb algebra, described in terms of certain diagrams. This in turn may be applied to give a complete analysis of the tilting modules at roots of unity using cellular theory. This is joint work with Henning Andersen and Ruibin Zhang.

Friday, 30 May 2014

• 14:15, Lecture Hall H8
  Michael Ehrig (Cologne): Three dualities: Schur-Weyl, skew Howe, Koszul
  Abstract: The talk relates classical dualities, like Schur-Weyl or skew Howe, to categorification methods. The BGG category O and various Khovanov type diagram algebras play an important role here. Furthermore the idea of varying the categorical objects to obtain new "classical" dualities will be explored. This is joint work with Catharina Stroppel.

Wednesday, 28 May 2014

• 14:15, Lecture Hall X-E0-202
  Colin Ingalls (Fredericton): Rationality of the Brauer-Severi Varieties of Sklyanin algebras
  Abstract: Iskovskih's conjecture states that a conic bundle over a surface is rational if and only if the surface has a pencil of rational curves which meet the discriminant in 3 or fewer points, (with one exceptional case). We generalize Iskovskih's proof that such conic bundles are rational, to the case of projective space bundles of higher dimension. The proof involves maximal orders and toric geometry. As a corollary we show that the Brauer-Severi variety of a Sklyanin algebra is rational.

Friday, 16 May 2014

• Colloquium in honour of the 75th birthday of Helmut Lenzing
  10:00, Room V2-210/216
  Osamu Iyama (Nagoya): Cohen-Macaulay representations of Geigle-Lenzing complete intersections
  Abstract: As a generalization of weighted projective lines, we introduce a class of commutative rings R graded by abelian groups L, which we call Geigle-Lenzing complete intersections. We will study L-graded Cohen-Macaulay R-modules, and show that there always exists a tilting object in the stable category. As an application we study when (R,L) is d-representation finite in the sense of higher dimensional Auslander-Reiten theory. This is a joint work with Herschend, Minamoto and Oppermann.
• Colloquium in honour of the 75th birthday of Helmut Lenzing
  11:30, Room V2-210/216
  Steffen Oppermann (Trondheim): Recollements for Geigle-Lenzing weighted projective varieties
  Abstract: In my talk I will discuss Geigle-Lenzing weighted projective varieties, which generalize Geigle and Lenzing's weighted projective lines to higher dimensional situations.
  First I will investigate how to "reduce weights" in the case of Geigle-Lenzing weighted P^n. It will turn out that reducing a weight by one will give rise to a recollement between three categories of coherent sheaves on GL-projective spaces, where the "quotient" is given by a hyperplane in the original space.
  I will then proceed to investigate the converse construction, that is I will build a new abelian category as the middle term of a recollement. It will turn out that this converse construction does not depend too much on the original varieties, so that it gives a method to introduce Geigle-Lenzing type weights also on hypersurfaces in other projective varieties.
• Colloquium in honour of the 75th birthday of Helmut Lenzing
  14:00, Room V2-210/216
  Atsushi Takahashi (Osaka): Weighted projective lines and extended cuspidal Weyl groups
  Abstract: We report on our recent study on a correspondence among weighted projective lines, cusp singularities and cuspidal Weyl groups. In particular, we discuss an isomorphism of Frobenius manifolds between the one from the Gromov-Witten theory for a weighted projective line and the one associated to the invariant theory of an extended cuspidal Weyl group.
• Colloquium in honour of the 75th birthday of Helmut Lenzing
  15:15, Room V2-210/216
  Dirk Kussin (Chemnitz): On tubes and multiplicities
  Abstract: We study three certain natural numbers associated with a tube of regular representations of a tame hereditary (or canonical) algebra over a perfect field. We present a general formula with these numbers and discuss implications for the Auslander-Reiten translation.
• Colloquium in honour of the 75th birthday of Helmut Lenzing
  16:45, Room V2-210/216
  Idun Reiten (Trondheim): My work with Helmut
  Abstract: I have written two joint papers with Helmut, one on hereditary categories over arbitrary fields, and one on generalizations of additive functions. I will discuss some of the results and the relationship to other work, especially for the first paper.

Thursday, 15 May 2014

• Colloquium in honour of the 75th birthday of Helmut Lenzing
  17:15, Room V2-210/216
  Ragnar-Olaf Buchweitz (Toronto): Matrix Factorizations over Elliptic Curves
  Abstract: Given a nonzero polynomial P in n variables, a matrix factorization of P consists of a pair of square matrices A,B of same size with entries from the polynomial ring such that AB = P Id, where Id stands for the appropriate identity matrix. If the polynomial is homogeneous one might further require that the entries of the matrices are homogeneous as well.
  A fundamental result by Orlov implies as a special case that equivalence classes of such homogeneous matrix factorizations for a cubic polynomial that defines an elliptic curve in the projective plane are in a natural, though still largely mysterious bijection with the isomorphism classes of indecomposable objects in the derived category of coherent sheaves on that elliptic curve. The structure of the latter is known since Atiyah's classification of such sheaves in 1957.
  After recalling the background just described, I will present results by my student Sasha Pavlov who uses this machinery to determine all possibilities for the degrees and sizes of the entries of such matrix factorizations and how this will enable us to find all such matrix factorizations eventually.
  Time permitting, I will outline how these results generalize to cones over elliptic curves embedded in higher projective spaces on the one hand and how, on the other, they relate to work of Lenzing et al. on weighted projective lines.

Wednesday, 14 May 2014

• 10:15, Room V2-210
  Ragnar-Olaf Buchweitz (Toronto): Higher Representation-Infinite Algebras from Geometric Tilting Objects
  Abstract: We will report on joint work with Lutz Hille on the recent notion of higher representation-infinite algebras. We show that a tilting object in a triangulated category of geometric dimension d, a notion proposed by Bondal, has an endomorphism ring that is higher representation-infinite if, and only if, it pulls back to a tilting object on the virtual affine canonical bundle over that category if, and only if, the endomorphism algebra has minimal global dimension, equal to d, and the tilting object is sheaf-like.
  The endomorphism ring of the pullback then yields the corresponding higher preprojective algebra. This proves, for example, that any full cyclic strongly exceptional sequence, a notion due to Hille-Perling that comprises the classical notion of a helix, gives rise to such pairs of d-representation-infinite algebras and their accompanying higher (d+1)-preprojective algebras, thereby providing plenty of examples.
  Intriguingly, such algebras also arise on non-Fano varieties, such as the second Hirzebruch surface or some non-isolated quotient singularities defined by abelian subgroups of special linear groups.
  Modulo an outstanding conjecture on the coherence of higher preprojective algebras and results of Minamoto, it follows that representation--infinite algebras are precisely the ones arising as endomorphism rings of minimal global dimension of sheaf-like tilting objects in triangulated categories of geometric dimension d.
  

Friday, 09 May 2014

• 13:15, Lecture Hall H8
  Andrew Hubery (Bielefeld): Realising projective schemes as Grassmannians of submodules
  Abstract: It has been known for some time that every projective variety over an algebraically-closed field can be realised as a Grassmannian of submodules for the Beilinson algebra. Using that every Grassmannian has a natural (in general non-reduced) scheme structure, we show that essentially the same construction provides a realisation of every projective scheme over an arbitrary field. The method of proof is a little different from previous work, since the natural scheme structure on the Grassmannian comes from its construction as a principal bundle for an action of the general linear group on the scheme of representations of the Beilinson algebra.

Friday, 25 April 2014

• 14:00, Lecture Hall H8
  Zhi-Wei Li (Bielefeld): A note on (co)slice model categories
  Abstract: There are various adjunctions between coslice and slice categories. We characterize when these adjunctions are Quillen equivalences. As an application, a triangle equivalence between the stable category of a Frobenius category and the homotopy category of a non-pointed model category is given.

Friday, 11 April 2014

• 14:00, Lecture Hall H8
  Moritz Groth (Nijmegen): Tilting Theory via Stable Homotopy Theory
  Abstract: Tilting theory is a derived version of Morita theory. In the context of quivers Q and Q' and a field k, this ammounts to looking for conditions which guarantee that the derived categories of the path algebras D(kQ) and D(kQ') are equivalent as triangulated categories.
  
  In this project (which is j/w Jan Stovicek) we take a different approach to tilting theory and show that some aspects of it are formal consequences of stability. Slightly more precisely, we show that certain tilting equivalences can be lifted to the context of arbitrary stable derivators. Plugging in specific examples this tells us that refined versions of these tilting results are also valid over arbitrary ground rings, for quasi-coherent modules on a scheme, in the differential-graded context, and also in the spectral context.
• 15:15, Lecture Hall H8
  Luke Wolcott (Appleton, Wisconsin): A topological localizing subcategory that isn't a Bousfield class
  Abstract: The stable homotopy category of spectra is a nice compactly generated tensor triangulated category. In this talk I'll look at a quotient of this category, the HFp-local category. I will calculate the Bousfield lattice, and give an example of a localizing subcategory that isn't a Bousfield class. Almost all the methods used make sense in other tensor triangulated categories, so anyone familiar with this concept should be able to follow.

Friday, 21 March 2014

• 11:15, Lecture Hall H11
  Patrick Wegener (Bielefeld): Transitive Hurwitz action on factorizations of Coxeter elements
  Abstract: We provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. Connections to representation theory of algebras arise from noncrossing partitions and exceptional sequences. (Joint work with B. Baumeister, M. Dyer and C. Stump.)
• 14:00, Lecture Hall H11
  Hiroyuki Minamoto (Sakai): Derived bi-commutator rings and derived completion
  Abstract: I will discuss (universal) properties of derived bi-commutator rings. The concept is not the same as in the classical case. But in nice cases we have the same results as in the classical case.
• 15:15, Lecture Hall H11
  Lutz Hille (Münster): On the derived category in global dimension two and matrix problems
  Abstract: jt. w. David Ploog
  In this talk we consider objects in the derived category and describe them using matrix problems. Roughly spoken we associate to any complex its cohomology and elements in the second extension group. In global dimension two we obtain an equivalence between additive categories, where we provide the derived catgeory with a new class of morphisms.
  The principal aim of the talk is to define the corresponding functor and to obtain criteria, when the functor becomes an equivalence. The construction works perfectly in global dimension two, however we also get partial results in higher global dimension.
  In a final part we give several applications of the construction.
• 16:45, Lecture Hall H11
  Alexander Nenashev (Toronto): Homological algebra for pointed sets
  Abstract: According to a result of Barrett-Priddy-Quillen, the stable homotopy groups (of spheres) are isomorphic to the K-groups of the category of finite pointed sets regarded as a Waldhausen category. The objective of my work is to approach the calculation of these K-groups in a way analogous to the work of Grayson who has calculated the K-groups of an exact category in terms of acyclic binary complexes. In its turn, the work of Grayson is based on the results of Thomason who works in terms of complexes, applying all the homological algebra available for an exact or abelian category. The category of finite pointed sets is not exact, even not additive (we do not have addition of morphisms), therefore we cannot apply the usual homological algebra to it. The objective of this talk is to develop a kind of homological algebra for pointed sets, which should rather be called the homological set theory.

Wednesday, 05 March 2014

• 10:15, Lecture Hall H11
  Benjamin Antieau (Seattle): Derived categories of genus one curves
  Abstract: I will discuss the problem of determining when two genus one curves have equivalent derived categories. When one of the curves has a rational point, then it is known that derived equivalence implies that the curves are isomorphic. When neither curve has a rational point, the problem is more subtle. I will explain how Danny Krashen, Matthew Ward, and myself use the twisted Brauer space to find a solution.
• 11:30, Lecture Hall H11
  Dirk Kussin (Chemnitz): Large tilting sheaves over tubular curves
  Abstract: We classify all large tilting sheaves in the tubular case and show that all these sheaves have a slope. (Joint work with Lidia Angeleri.)

Friday, 14 February 2014

• 14:15, Lecture Hall H10
  Hugh Thomas (New Brunswick): Combinatorics of AR-translation in finite type cluster categories, and analogues
  Abstract: A cluster category of Dynkin type admits a finite cyclic action given by the Auslander-Reiten translation, which induces an action on the set of cluster-tilting objects. I will discuss an alternative way to realize this action by means of a walk on the exchange graph of cluster tilting objects. There is another set of objects associated to the Dynkin type, called the "non-nesting partitions", which are equinumerous with the cluster-tilting objects. There is an analogous cylic action on the non-nesting partitions. Conjecturally, there are natural bijections between the two sets which intertwine these two actions, but the non-nesting side of the story is not well understood. In particular, a useful representation-theoretic understanding of the non-nesting partitions is still lacking. This talk is based on part of a joint project with Christian Stump and Nathan Williams.

Tuesday, 11 February 2014

• 10:00, Room V2-210/216
  Julia Sauter (Bielefeld): The Auslander bijections: Introduction and examples
  Abstract: This talk is an introduction of Auslander’s old approach to dissect mod-A for an artin algebra A with lattices [-> Y) and C-[-> Y) of equivalence classes of morphisms ending in a fixed A-module Y. For C-[->Y) this depends on the choice of another module C. The Auslander bijection identifies C-[-> Y) with the lattice of End(C)-submodules of Hom (C,Y). Following Ringel's survey, we explain examples and properties of these lattices. If A is a finite-dimensional K-algebra over an algebraically closed field K, then C-[-> Y ) can be identified with (a union of) quiver Grassmannians.
• 11:15, Room V2-210/216
  Henning Krause (Bielefeld): The Auslander bijections: Functors determined by objects revisited
  Abstract: The Auslander bijections are based on morphisms determined by objects; they are best understood by looking at the more general concept of a functor determined by an object. This was the ingenious insight of Auslander which led him to his 1978 Philadelphia notes. In my talk I carry these ideas a bit further and give a new (and rather elementary) proof of one of Auslander's main results from these notes. I will end with some open problems.
• 14:00, Room V2-210/216
  Hugh Thomas (New Brunswick): Cofinite quotient-closed subcategories of quiver representations (plus...)
  Abstract: Let Q be a quiver without oriented cycles, and k an algebraically closed field. We say that a subcategory of kQ-mod is cofinite if it is full and contains all but finitely many of the indecomposables. We show that the cofinite quotient-closed subcategories of kQ-mod are naturally in bijection with the elements of the Weyl group associated to Q. We also extend this result to hereditary algebras which are finite-dimensional over a finite field, using Frobenius twisting, which I will explain. This talk is based on a joint paper with Steffen Oppermann and Idun Reiten, arXiv:1205.3268.
• 15:15, Room V2-210/216
  Osamu Iyama (Nagoya): Stable categories of one-dimensional hypersurface singularities
  Abstract: We show that the stable categories of graded Cohen-Macaulay modules over one-dimensional hypersurface singularities k[x,y]/(f) with standard grading have tilting objects U. Moreover we show that their endomorphism algebras are 2-representation finite, and provide us with a family of selfinjective quivers with potential. This is a joint work with Araya, Herschend, Takahashi and Yamaura.
• 16:45, Room V2-210/216
  Idun Reiten (Trondheim): Some aspects of tau-tilting theory
  Abstract: The theory of tau-tilting was initiated and developed in a paper with Adachi and Iyama. We start with recalling some basic definitions and results. Then we concentrate on discussing how torsion classes are used to show that almost complete support tau-tilting modules have exactly two complements, and how a left/right duality result is used for computing one complement from the other one.

Friday, 31 January 2014

• 14:15, Lecture Hall H10
  Sebastian Klein (Utrecht): Chow groups and intersection product for algebraic tensor triangulated categories
  Abstract: We define Chow groups for tensor triangulated categories and present some calculations from algebraic geometry and modular representation theory. We then indicate how to construct an intersection product for these Chow groups, if the category is algebraic and satisfies an analogue of the Gersten conjecture from algebraic K-theory. By a result of Grayson, this product generalizes the usual intersection product on a non-singular algebraic variety.

Friday, 24 January 2014

• 13:15, Room V2-210
  Greg Stevenson (Bielefeld): Derived categories of quivers over Noetherian rings
  Abstract: Given a Noetherian ring R, or a Dynkin quiver Q and a field k, one can consider the unbounded derived categories D(R) and D(kQ), of R-modules, and of representations of the path algebra kQ, respectively. These triangulated categories are somewhat well understood in the sense that one has, in both cases, a full classification of the localising subcategories. It's thus natural to ask if one can combine these classification results to say something about D(RQ), the unbounded derived category of the R-linear path algebra. I'll discuss joint work with Ben Antieau which shows that not only can one combine these classifications, but that it is possible to reduce such classification problems to the case of fields in a quite general setting.
• 14:30, Room V2-210
  Olaf Schnürer (Bonn): Some enhancements of categories of coherent sheaves and applications
  Abstract: We introduce generalizations of Cech enhancements for the bounded derived category D^b(Coh(X)) of coherent sheaves on a suitable scheme X and for its subcategory Perf(X) of perfect complexes. These enhancements are especially well suited for addressing the following questions: (1) homological smoothness of D^b(Coh(X)); (2) relation between homological smoothness of Perf(X) and geometric smoothness of X; (3) translation of Fourier-Mukai transformations to the world of dg algebras.

Friday, 17 January 2014

• 14:15, Lecture Hall H10
  Julia Sauter (Leeds): Type A-Springer theory and quiver flag varieties
  Abstract: We explain how to construct the positive half of the quantum group of a quiver from a single bifunctor F by using a geometric construction which we call Springer theory. Springer theory gives us a multiplicative family of graded algebras called Steinberg algebras, here these are the KLR-algebras. The Grothendieck group of graded projective modules over them is the positive half of the quantum group. A special class of modules over the Steinberg algebras is given by (co)homology groups of quiver flag varieties. For the A_n-equioriented quiver case I found a cell decomposition of the quiver flag varieties which is a geometric realization of a cellular algebra structure of this KLR-algebra. In my phd thesis I worked on partial generalizations to other reductive groups, but for this talk we restrict to type A groups.

Friday, 10 January 2014

• 14:00, Lecture Hall H10
  Reiner Hermann (Bielefeld): Non-commutative Poisson structures for quasitriangular Hopf algebras
  Abstract: Let A be an associative and unital algebra over a commutative ring. By definition, a non-commutative Poisson structure for A is an element in the second Hochschild cohomology group of A, such that its Gerstenhaber square bracket vanishes. In this talk, we will explain how each degree-2-element inside the cohomology ring of a quasitriangular Hopf algebra H gives rise to a non-commutative Poisson structure for H. Our approach heavily relies on a generelization of Stefan Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology.
• 15:15, Lecture Hall H10
  Yu Zhou (Bielefeld): Cluster categories from marked surfaces with punctures
  Abstract: A marked surface S is an oriented compact surface with marked points. To any triangulation of a marked surface, Fomin, Shapiro, Thurston and Labardini-Fragoso associated a quiver with potential (Q, W). By Amiot's work, there is a 2-Calabi-Yau triangulated category C with cluster tilting object T such that the quotient category C/T is isomorphic to the module category of the Jacobian algebra of (Q,W). In the case when S has punctures and non-empty boundaries, we show that there is a bijection between tagged curves in S and string objects in C which does not depend on the choice of the triangulations. Under this bijection, we interpret dimensions of Ext's as intersection numbers, the Calabi-Yau reduction as cutting and the Auslander-Reiten translation on string objects as tagged rotation. Moreover the cluster exchange graphs in such cases are shown to be connected. This is a joint work with Yu Qiu.
  

Wednesday, 18 December 2013

• 10:15, Room V5-227
  Kay Großblotekamp (Paderborn): Mengenwertige Darstellungen des Kroneckerköchers
  Abstract: Mengenwertige und lineare Darstellungen des Kroneckerköchers werden verglichen. Die Kategorie der mengenwertigen Darstellungen ist isomorph zur Kategorie der Köcher. Es gibt damit einen Vergißfunktor V, der lineare Darstellungen als Köcher interpretiert. Die Objekte im Bild dieses Funktors werden durch die unzerlegbaren linearen Darstellungen bestimmt. Wir geben eine graphentheoretische Beschreibung der Köcher, die V den unzerlegbaren linearen Darstellungen zuordnet (für endliche Grundkörper). Einige Eigenschaften des Linksadjungierten von V werden untersucht. Unter anderem lässt sich jeder Köcher in ein Produkt von Köchern einbetten, deren Zusammenhangskomponenten aus einer von sechs Klassen stammen.

Friday, 13 December 2013

• Workshop "Noncommutative resolutions of singularities" (Seminar Algebraic Geometry and Representation Theory)
  10:15, Room V2-210/216

Thursday, 12 December 2013

• Faculty Colloquium
  17:15, Room V2-210/216
  Markus Reineke (Wuppertal): Wall-crossing formulas
  Abstract: Wall-crossing formulas arose in the last few years in string theory and algebraic geometry, most notably in the work of Kontsevich and Soibelman on motivic Donaldson-Thomas invariants. The aim of the talk is to give a non-technical introduction to the ideas leading to wall-crossing formulas, and to show some of their applications, for example to dilogarithm identities, hypergeometric functions, the enumeration of plane curves, and certain new zeta functions.

Wednesday, 11 December 2013

• 10:15, Room V3-201
  Atsushi Takahashi (Osaka): Weyl Groups and Artin Groups Associated to Weighted Projective Lines
  Abstract: After explaining our motivation coming from mirror symmetry, we report on our recent study of a correspondence among weighted projective lines, cusp singularities and cuspidal root systems. A conjectual relation among Weyl groups, Artin groups and the spaces of Bridgeland's stability conditions for some triangulated categories for weighted projective lines will also be explained.

Friday, 06 December 2013

• 13:15, Room V2-210
  Philipp Lampe (Bielefeld): Divisor class groups of cluster algebras
  Abstract: We wish to present a necessary and sufficient and computer-checkable criterion for an acyclic cluster algebra to be a unique factorization domain. The proof relies on its divisor class group. As an illustration, we use the criterion to classify cluster algebras that are simultaneously unique factorization domains and of finite type. Moreover, we compute divisor class groups of some cluster algebras of infinite type.
• 14:30, Room V2-210
  Andrew Hubery (Bielefeld): Subgroups of pointed Weyl groups, and non-crossing partitions
  Abstract: Starting from a lattice with a non-degenerate bilinear from, and a complete exceptional sequence, we can associate a Weyl group and a Coxeter element, and hence a poset of non-crossing partitions. We show that the Weyl group induces a functor from a category of such lattices to the category of (pointed) groups, and that the poset of subgroups is isomorphic to the poset of non-crossing partitions. We therefore obtain interesting monomorphisms of Weyl groups where the image is in general not a parabolic subgroup.
• 16:00, Room V2-210
  Christof Geiss (Mexico City): On the uniqueness of non-degenerate potentials for mutation finite quivers
  Abstract: Recall that by a result of Felikson-Shapiro-Tumarkin a connected quiver with at least 3 vertices is mutation finite if and only if it comes from a triangulation of a surface or if it belongs to a list of 11 exceptional mutation classes. We say that a mutation finite quiver is critical if it belongs to the following list:
  C0) Q comes from the triangulation of a torus with on boundary component and one marked point on the boundary;
  C1) Q comes from the triangulation of a closed surface of positive genus with 1 puncture;
  C2) Q comes from the triangulation of a closed surface of positive genus with 2 punctures;
  C3) Q is mutation equivalent to the quiver X_7 (Derksen-Owen);
  C4) Q comes from the triangulation of a closed sphere with 4 punctures.
  Theorem. Let Q be a mutation finite quiver which is not critical, then there is for Q up to (weak) right equivalence only one non-degenerate potential.
  Remarks: (1) In the cases C0), C1) and C4) there are definitively several (weak) right equivalence classes of non-degenerate potentials. The main open case is C2).
  (2) Non-uniqueness of non-degenerate potentials seems to be related to unpleasent behaviour of the corresponding cluster algebras.
  (3) If time permits we discuss the representation types of the corresponding Jacobian algebras.
  This is a report on joint work with D. Labardini-Fragoso and J. Schröer.

Friday, 29 November 2013

• 13:15, Lecture Hall H10
  Dan Zacharia (Syracuse): The extended degree zero subalgebra of the ext algebra of a linear module
  Abstract: I will talk on joint work with Ed Green and Nicole Snashall. Let k be a field and let R be a Koszul k-algebra. Let M be a linear k-module and let Γ be the ext-algebra of M, that is Γ = Ext_R*(M,M). View Γ as a bigraded algebra with the bigrading induced by the homological degree and by the internal grading of M. We consider the extended degree zero subalgebra (for lack of a better name) ∆M = Ext_R*(M,M)_0 of Γ. It turns out that the extended degree zero subalgebra can be used to obtain a characterization of the graded center of a Koszul algebra. I will also present some other applications of the ideas involved.
• 14:30, Lecture Hall H10
  Otto Kerner (Düsseldorf): Torsion classes for wild hereditary algebras
  Abstract: In my talk I will present the following result: If H is a connected finite dimensional wild hereditary algebra, then there exist infinitely many torsion classes in H-mod without nonzero Ext-projective modules. This answers a question of Adam-Christiaan von Roosmalen.

Friday, 15 November 2013

• 14:15, Room V2-210/216
  Fajar Yuliawan (Bielefeld): Homotopy category of an algebra with radical square zero and Leavitt path algebras
  Abstract: For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential. A related, but different, result for the homotopy category of acyclic complexes of projective modules is also given. These equivalences follow from a version of Koszul duality for algebras with radical square zero and a triangle equivalence induced by a graded universal localization from the path algebra of a quiver to the corresponding Leavitt path algebra. The talk is based on a recent paper arXiv:1301.0195 by Xiao-Wu Chen and Dong Yang.

Saturday, 09 November 2013

• Workshop "Polynomial Functors and Schur Algebras"
  09:00, Room V2-210/216

Friday, 08 November 2013

• Workshop "Polynomial Functors and Schur Algebras"
  10:00, Room V2-210/216

Thursday, 07 November 2013

• Workshop "Polynomial Functors and Schur Algebras"
  10:00, Room V2-210/216
• Faculty Colloquium
  17:15, Room V2-210/216
  Catharina Stroppel (Bonn): From classical Schur-Weyl duality to quantized skew Howe dualities
  Abstract: Classical Schur-Weyl duality connects the representation theory of the general linear group with the symmetric group and goes back to Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups. A quantization of the duality plays an important role in basic knot theory and yields for instance the famous Jones polynomial. The talk will start from the basic construction and explain interesting generalizations. The tricky point here is the ambiguous role played by the symmetric group as a centralizer group as well as a Weyl group. This leaves us with questions like: what are good generalizations? Are there categorical or geometric interpretations of such dualities? Why is it difficult to quantize such generalization? And why should we care?

Friday, 25 October 2013

• 14:00, Room U2-113
  Jerzy Weyman (Storrs, Connecticut): Local cohomology supported in determinantal varieties
  Abstract: Let K be a field of characteristic zero. Consider the polynomial ring S over K on the entries of a generic m x n matrix X. Let I be the ideal in S generated by p x p minors of X. I explain how to calculate completely the local cohomology modules of S with respect to I. I will also explain why the problem is interesting. It turns put the result allows to classify the maximal Cohen-Macaulay modules of covariants for the action of SL(n) on the set of m n-vectors. It also allows to describe the equivariant simple D-modules, where D is the Weyl algebra of differential operators on the space of m x n matrices. This is a joint work with Claudiu Raicu and Emily Witt. The relevant references are arXiv:1305.1719 and arXiv:1309.0617.
• 15:15, Room U2-113
  Yuriy Drozd (Kiev): Tilting and resolutions for singular curves
  Abstract: For singular projective curves we construct a full embedding of the derived category of quasi-coherent sheaves into the derived category of modules over a finite dimensional algebras of finite global dimension. Properties of this embedding are studied. We also consider analogous construction for non-commutative curves. (This is a joint work with I.Burban and V.Gavran.)

Friday, 18 October 2013

• 14:00, Room U2-113
  Greg Stevenson (Bielefeld): Graded modules versus dg-modules
  Abstract: Given a graded ring one can naturally associate to it two different derived categories. On one hand there is the derived category of graded modules and on the other hand one can consider the graded ring as a dg-algebra with trivial differential and form the derived category of dg-modules. The aim of the talk is to explain a connection between the structure of these two triangulated categories. In the noetherian case this unifies previous work on classifying localising subcategories by the speaker and Dell'Ambrogio in the case of graded modules and Benson, Iyengar, and Krause in the case of dg-modules.
• 15:15, Room U2-113
  Jeanne Scott (Chennai): The twist, again - lecture two
  Abstract: In this talk, which is a report of joint work with R. Marsh, I will explain how to evaluate Laurent expansions for twisted Plücker coordinates with respect to any seed of the Grassmannian arising from a special class of planar networks called Postnikov diagrams. I will show how these expansions, which are predicted using the theory of cluster algebras, can be explicitly tabulated using perfect matchings within a bipartite graph dual to the Postnikov diagram.
  

Friday, 20 September 2013

• 14:00, Room V3-204
  Jeanne Scott (Chennai): The twist, by matchings
  Abstract: The Grassmannian 'twist' is a birational automorphism of the type-A Grassmannian introduced by Berenstein and Zelevinsky, albeit in a much wider context, as a tool to study factorisations of elements within strata of the unipotent radical of a complex semi-simple algebraic group by Chevalley generators. It has the property that, up to coefficients, the twist of a cluster variable (within the homogeneous coordinate ring of the Grassmannian) is a cluster variable; moreover the twist preserves compatibility between any two cluster variables. In this talk, which is a report of joint work with R. Marsh, I will explain how to evaluate Laurent expansions for twisted Pl\"ucker coordinates with respect to any seed of the Grassmannian arising from a special class of planar networks called Postnikov diagrams. I will show how these expansions, which are predicted using the theory of cluster algebras, can be explicitly tabulated using perfect matchings within a bipartite graph dual to the Postnikov diagram.
• 15:15, Room V3-204
  Robert Marsh (Leeds): Dimer models with boundary and cluster categories associated to Grassmannians
  Abstract: Joint work with K. Baur (Graz) and A. King (Bath).
  A dimer model can be defined as a quiver embedded into a surface in such a way that the complement is a disjoint union of disks with oriented boundaries. Such models can also be considered in the case of a surface with boundary. The Postnikov diagrams used by J. Scott to describe the cluster structure of the homogeneous coordinate ring of the Grassmannian give rise to dimer models on a disk in this sense.
  We associate a natural algebra to such a dimer model. This algebra is a modified version of the corresponding Jacobian algebra, taking the boundary into account. Taking the sum of the idempotents corresponding to boundary vertices, we obtain an idempotent subalgebra, which we call the boundary algebra. We show that it is independent of the choice of dimer model and coincides with an algebra that B. Jensen, A. King and X. Su have used to model the cluster structure of the homogeneous coordinate ring of the Grassmannian categorically.
  

Wednesday, 24 July 2013

• 13:15, Lecture Hall H9
  Chrysostomos Psaroudakis (Ioannina): Gorenstein Artin algebras arising from Morita contexts
  Abstract: Associated to any Morita context M there is an associative ring, called the Morita ring of M. In this talk we discuss Morita rings concentrating mainly at representation-theoretic and homological aspects. This is joint work with Edward L. Green.
• 14:30, Lecture Hall H9
  Ryan Kinser (Boston): Type A quiver loci and Kazhdan-Lusztig varieties
  Abstract: We show how to embed a representation variety of a type A quiver into a Kazhdan-Lusztig variety (Schubert variety intersected with opposite Schubert cell). The embedding takes orbit closures to Schubert varieties intersected with the opposite cell. The talk will be example based, requiring no previous knowledge of Schubert varieties.
  This has implications for the geometry of the orbit closures, such as recovering a theorem of Zwara and Bobiński that the orbit closures are normal and Cohen-Macaulay, and also leads to formulas for cohomology and K-classes of the orbit closures.
• 16:00, Lecture Hall H9
  Grzegorz Bobinski (Torun): Moduli spaces for quasi-tilted algebras
  Abstract: Weyman has formulated a conjecture that an algebra is tame if and only if all King's moduli spaces are products are projective spaces. Efforts in order to verify this conjecture has been made Chindris. In particular, he has showed that wild quasi-tilted algebras always have a singular moduli space. During my talk I will discuss the proof of the other implication in the case of quasi-tilted algebras.

Friday, 19 July 2013

• Workshop "McKay Correspondence" (Seminar Algebraic Geometry and Representation Theory)
  10:15, Room V2-210/216

Friday, 21 June 2013

• 13:15, Room V2-213
  Sefi Ladkani (Bonn): Jacobian algebras from closed surfaces, derived equivalences and Brauer graph algebras
  Abstract: To any ideal triangulation of a surface with marked points Labardini-Fragoso has associated a quiver with potential, thus linking the work of Fomin, Shapiro and Thurston on cluster algebras arising from marked surfaces with the theory of quivers with potentials and their mutations initiated by Derksen, Weyman and Zelevinsky.
  We show that for any surface without boundary, the associated quivers with potentials are not rigid and their (completed) Jacobian algebras are finite-dimensional, symmetric and derived equivalent. This settles a question that has been open for some time and also provides an explicit construction of infinitely many families of finite-dimensional symmetric Jacobian algebras. Moreover, these Jacobian algebras are closely related to Brauer graph algebras arising naturally from triangulations of the surface.
• 14:30, Room V2-213
  Hideto Asashiba (Shizuoka): Lax functors of bicategories and derived equivalences with application to triangular matrix algebras
  Abstract: Let k be a field. We first review a gluing process of derived equivalences of k-categories using Grothendieck constructions of colax functors from a small category I to the 2-category of k-categories. Next we discuss a generalization of this process by extending the definition of Grothendieck constructions to those of lax functors from I to the bicategory of k-categories and bimodules over them to recover triangular matrix algebras (or more generally tensor algebras of k-species).

Friday, 14 June 2013

• Workshop "McKay Correspondence" (Seminar Algebraic Geometry and Representation Theory)
  10:15, Room V2-210/216

Saturday, 08 June 2013

• Workshop "McKay Correspondence" (Seminar Algebraic Geometry and Representation Theory)
  10:15, Room V2-210/216

Friday, 07 June 2013

• 13:15, Room V2-213
  Adam-Christiaan van Roosmalen (Berkeley): Cluster categories associated to new hereditary categories
  Abstract: This is joint work with Jan Šťovíček. Given a finite quiver, one can associate a cluster category by considering orbits of the bounded derived category of finite dimensional representations. In this talk, we want to replace the original quiver by a suitable small category such that the orbit construction still makes sense, thus obtaining new examples of 2-Calabi-Yau categories with cluster tilting subcategories. We will consider some examples where one can use combinatorics to describe the cluster tilting subcategories, as is done by Holm and Jørgensen in the case of the infinite Dynkin quiver A_infinity using triangulations of the infinity-gon.
• 14:30, Room V2-213
  Julia Worch (Kiel): Module categories and Auslander-Reiten theory for generalized Beilinson algebras
  Abstract: Inspired by the work of Carlson, Friedlander, Pevtsova and Suslin in the modular representation theory of finite group schemes, we introduce the categories of modules of constant Jordan type and modules with the equal images property for generalized Beilinson algebra. We give a homological characterization of these subcategories which enables us to apply general methods from Auslander-Reiten theory and thereby obtain information concerning the occurrence of the corresponding modules within the Auslander-Reiten quiver of the Beilinson algebra.

Friday, 31 May 2013

• 13:15, Room V2-213
  Julia Sauter (Leeds): From Springer theory to monoidal categories
  Abstract: To every Springer Theory one gets a category of (shifts of) perverse sheaves generated by direct summands in the BBD decomposition theorem, which we call Lusztig's perverse sheaves. When one chooses "additive" families of Springer theories, it is possible to define a convolution product on the associated category of perverse sheaves. This way one gets a monoidal category, the homomorphisms are given by Steinberg algebras. Lusztig proved that when one starts with quiver-graded Springer theory, then this gives a monoidal categorification of the positive half of the quantum group associated to the quiver. We study this construction for more general Springer theories and explain the example of symplectic quiver-graded Springer theory.
• 14:30, Room V2-213
  Michael Cuntz (Kaiserslautern): Weyl groupoids and arrangements
  Abstract: The Weyl groupoid is a symmetry structure which was originally introduced as an invariant of Nichols algebras. The classification of finite Weyl groupoids revealed further applications in geometry and combinatorics. In this talk we will see this connection for more general Cartan schemes, and discuss the recently initiated classification of affine Weyl groupoids.
• 16:00, Room V2-213
  Jose Antonio de la Pena (Guanajuato): On the Mahler measure of Coxeter polynomials
  Abstract: Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected and triangular, hence of finite global dimension. We say that A is of cyclotomic type if the characteristic polynomial p(x) of of the Coxeter transformation is a product of cyclotomic polynomials, equivalently, if the Mahler measure  M(p)=1. We consider the many examples of algebras of cyclotomic type in the representation theory literature and show some common properties. We also consider algebras not of cyclotomic type with small Mahler measure of their Coxeter polynomial. In 1933, D. H. Lehmer  found that the polynomial T^{10} + T^9 - T^7 - T^6 - T^5 - T^4 - T^3 + T + 1 has Mahler measure m = 1.176280..., and he asked if there exist any smaller values  exceeding 1. We prove that either M(p)=1 or M(p)≥m for strongly simply connected algebras A.

Friday, 24 May 2013

• 14:00, Room V2-213
  Julia Sauter (Leeds): Classical and quiver-graded Springer theory
  Abstract: We introduce Springer Theory as a geometric construction of (some) graded convolution algebras (Steinberg algebras) together with certain modules, called Springer fibre modules. The BBD-decomposition theorem gives a parametrization of (graded) indecomposable projectives and simple modules for the Steinberg algebras. The two main examples are classical and quiver-graded Springer theory. For the classical Springer Theory the Steinberg algebra is the group ring of the Weyl group (ass. to a reductive group) and the Springer correspondence identifies simple modules with isotypic subspaces of the Springer fibre modules. The Steinberg algebras for quiver-graded Springer theory are quiver Hecke algebras (=KLR-algebras) introduced by Khovanov-Lauda and Rouquier. My result here is an explicit calculation of Steinberg algebras in a more general setup.
  
• 15:15, Room V2-213
  Matthias Warkentin (Chemnitz): On the global structure of infinite mutation graphs
  Abstract: Let Q be an acyclic quiver and K an algebraically closed field. The exchange graph of tilting modules over KQ introduced by Riedtmann and Schofield has been studied extensively by Happel and Unger. After the introduction of cluster algebras and cluster categories it has been shown that this exchange graph can be seen as a part of the exchange graph of the cluster algebra given by Q, which is governed by the combinatorics of quiver mutations. We explain how elementary considerations about quiver mutations can be used to understand the structure of the corresponding exchange graphs. In particular, our results (combined with results by Felikson, Shapiro and Tumarkin) yield an "almost complete" answer to Unger's conjecture about the number of connected components.

Friday, 26 April 2013

• 13:15, Room V2-213
  Phillip Linke (Bielefeld): Computational approach to the Artinian conjecture (Part I)
  Abstract: What is generic representation theory? When looking at the category F=Func(mod F_q,Mod F_q) we obtain that a functor G in F generically gives rise to representations of GL(V) for all V in mod F_q. By the Yoneda-lemma we know how certain projectives in F look like. For each V in mod F_q, Hom(V,-) is projective. Such a projective is called a standard projective. It turns out that these standard projective even generate the whole category.
  In the 1980s Lionel Schwartz conjectured that all the standard projectives would be noetherian. If true this would imply that every finitely generated functor in F admits a projective resolution by finitely generated projectives. There are partial results that back up this conjecture but no solution so far.
  In the talk we will not reach quite as far. The aim is to give an idea why the category F is at least coherent. That means that every finitely presented functor admits a resolution by finitely generated projectives. To get to this goal we will use certain combinatorial properties of the dimension function phi(G,n)=dim_{F_q}G(F_{q^n}) for a functor G in F.
• 14:30, Room V2-213
  Phillip Linke (Bielefeld): Computational approach to the Artinian conjecture (Part II)
  Abstract: See Part I.
• 16:00, Room V2-213
  Sven Meinhardt (Wuppertal): Motivic DT-invariants of (-2)-curves
  Abstract: In the first part of my talk I will gently introduce (0,-2)-curves and sketch how they show up in resolutions of singular 3-folds. After that, an alternative non-commutative resolution using quivers with potential is given. Finally, I will briefly introduce Donaldson-Thomas invariants and state the answer in our situation which is the main result of a joint work with Ben Davison.

Friday, 19 April 2013

• 14:30, Room V2-213
  Martin Kalck (Bielefeld): Singularity categories of gentle algebras
  Abstract: We give an explicit description of the triangulated category of singularities (in the sense of Buchweitz and Orlov) for all finite dimensional gentle algebras. Examples include Jacobian algebras arising from triangulations of unpunctured marked Riemann surfaces and algebras which are derived equivalent to certain singular projective curves. Moreover, we recover part of a derived invariant for gentle algebras, which was discovered by Avella-Alaminos & Geiß.
  

Friday, 12 April 2013

• Workshop Noncrossing Partitions
  09:00, Room V2-210/216
  Claus Michael Ringel (Bielefeld): The non-crossing partitions for any Dynkin type are the antichains in the corresponding root poset. On antichains in posets and in additive categories
  Abstract: Dealing with simply laced Dynkin diagrams, Ingalls and Thomas (Compos. Math. 145, 2009) gave an interpretation of the set of non-crossing partitions in terms of the representation category of a Dynkin quiver: they exhibited, for example, a bijection between the non-crossing partitions and the wide subcategories or also the torsion classes. These results can be reformulated in terms of antichains in additive categories and extended to the non-simply laced cases B_n, C_n, F_4, G_2 and the corresponding hereditary abelian categories. We will show in which way the representation theory approach sheds light on the relationship between crossing and nesting; this relationship is well-known in the cases A_n, but seemed to be quite mysterious in the remaining cases.
• Workshop Noncrossing Partitions
  10:15, Room V2-210/216
  Friedrich Götze (Bielefeld): Free Probability and Noncrossing Partitions
• Workshop Noncrossing Partitions
  11:30, Room V2-210/216
  Patrick Wegener (Bielefeld): The dual braid monoid (after Bessis)
  Abstract: Considering a finite Coxeter system (W,S) one can construct a monoid structure for the associated Artin group, the so called classical braid monoid. Replacing (W,S) by (W,T,c), where T is the set of all reflections in W and c a Coxeter element, we construct a new monoid structure for the associated Artin group. Essential for the construction is the lattice of noncrossing partitions and some of its properties. Like the classical monoid this monoid will be Garside. This analogy indicates that there might be a "dual" way of studying Coxeter systems.
• Workshop Noncrossing Partitions
  14:00, Room V2-210/216
  Philipp Lampe (Bielefeld): Combinatorial models for cluster algebras via noncrossing partitions
  Abstract: A cluster algebra is a commutative ring together with a distinguished set of generators called cluster variables. We obtain the cluster variables from given initial variables by a very concise combinatorial mutation process. The cluster algebras with only finitely many cluster variables have been classified by finite type root systems. Here, cluster variables correspond to almost positive roots. In this talk, we wish to discuss which almost positive roots arise from the same cluster. We introduce several combinatorial models and bijections between clusters, Coxeter-sortable elements and noncrossing partitions.
• Workshop Noncrossing Partitions
  15:15, Room V2-210/216
  Henning Krause (Bielefeld): A K-theoretic study of exceptional sequences
  Abstract: Given a category of representations, we consider its Grothendieck group together with the Euler form (the bilinear form defined by the alternating sum of dimensions of Ext-spaces). In this setup, one defines roots, reflections, a Weyl group, and exceptional sequences. We associate to each exceptional sequence a Coxeter element (a product of reflections in the Weyl group). Under suitable assumptions, this yields a bijection between all exceptional sequences and the noncrossing partitions (viewed as elements of the Weyl group). In my talk, I'll explain this construction and discuss some examples.
• Workshop Noncrossing Partitions
  16:30, Room V2-210/216
  Gennadiy Chistyakov (Bielefeld): Distributions of commutators and anti-commutators

Thursday, 11 April 2013

• Workshop Noncrossing Partitions
  15:30, Room V2-210/216
  Christopher Voll (Bielefeld): Noncrossing partitions and Coxeter groups
  Abstract: I will explain some fundamental aspects of the lattice of noncrossing partitions of a general (finite) Coxeter group. The exposition will be almost self-contained; some familiarity with the basics of Coxeter group theory might help, but is not essential.

Friday, 25 January 2013

• 13:15, Lecture Hall H6
  Britta Späth (Kaiserslautern): An approach to global/local conjectures in the representation theory of finite groups
  Abstract: Much of the recent work in the representation theory of finite groups is centered around the global/local conjectures, notably the conjectures from Alperin, Brauer and McKay. An underlying idea of these conjectures is that certain aspects of the representation theory of a finite group should be determined "locally", that is, by the representation theory of so-called local subgroups (e.g., the normalisers of certain p-subgroups).
  In the talk I describe how these conjectures can be reduced to questions on simple groups. Furthermore I sketch in which cases these questions can be answered completely.
  
• 14:30, Lecture Hall H6
  Jan Schröer (Bonn): The representation type of Jacobian algebras
  Abstract: This is joint work with Christof Geiss and Daniel Labardini-Fragoso. We determine the representation type of (almost) all Jacobian algebras P(Q,S) arising from a 2-acyclic quiver Q and a non-degenerate potential S. Such algebras were introduced by Derksen, Weyman and Zelevinsky and play a central role in relating cluster algebras with the representation theory of quivers.
  
• 16:00, Lecture Hall H6
  Jon Carlson (Athens, Georgia): Modules of constant radical type and bundles on Grassmannians
  Abstract: This is joint work with Eric Friedlander and Julia Pevtsova. We introduce higher rank variations on the notion of $\pi$-points as defined by the second two authors for representations of finite group schemes. Using this we can define module of constant r-radical and r-socle type. Such modules determine bundles over the Grassmannian associated to the higher rank $\pi$-points in the case that the group scheme is infinitesimal of height one. When the group scheme is an elementary abelian p-group, there is universal function for computing the kernel bundles as modules over the structure sheaf of the Grassmannian of r-planes in n space. These ideas also extend to various sorts of subalgebra of restricted p-Lie algebras.

Friday, 18 January 2013

• 13:15, Lecture Hall H6
  Lennart Galinat (Köln): Orlov's Equivalence and Maximal Cohen-Macaulay Modules over the Cone of a (plane) Elliptic Curve
  Abstract: In 1982 Kahn showed that the category of MCM modules over a simple elliptic surface singularity is representation tame. However his description of its indecomposable objects is far from being explicit.
  In my talk I shall present a classification of all rank one matrix factorisations of a cone over a plane elliptic curve which is based on more recent techniques including Orlov's equivalence for graded MCMs.
  Moreover I shall explain a (computer algebra based) way to describe all indecomposable matrix factorisations for such singularities.
• 14:30, Lecture Hall H6
  Claus Michael Ringel (Bielefeld): From submodule categories to preprojective algebras
  Abstract: Let S(n) be the category of invariant subspaces of nilpotent operators with nilpotency index at most n. Such submodule categories have been studied already in 1936 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra of type A_n; the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroeer). Direct connections between the submodule category S(n) and the module category of the preprojective algebra of type A_{n-1} have been established quite a long time ago by Auslander and Reiten, and recently also by Li and Zhang, but apparently this remained unnoticed. The lecture is based on joint investigations with Zhang Pu and will provide details on this relationship. As a byproduct we see that here we deal with ideals I in triangulated categories T such that I is generated by an idempotent and T/I is abelian.
• 16:00, Lecture Hall H6
  Vanessa Miemietz (Norwich): The extension algebra of Weyl modules for GL_2
  Abstract: I will explain how to use a homological duality of 2-functors to give an explicit construction of the Ext-algebra of Weyl modules, which in particular yields a multiplicative (up to sign) basis.

Thursday, 17 January 2013

• 10:15, Room V5-227
  Vanessa Miemietz (Norwich): 2-functors and homological duality
  Abstract: We will explain how certain 2-functors encoding the rational representation theory of GL_2 in positive characteristic commute with homological dualities, which makes it possible to compute various extension algebras.

Friday, 21 December 2012

• 13:15, Lecture Hall H6
  Lutz Hille (Münster): On the irreducible components for algebras over double quivers
  Abstract: Several algebras defined by a double quiver with certain relations, like the preprojective algebra, are of geometric interest. In particular, the number of irreducible components of the corresponding representation space play an important role. One of the most prominent examples is the construction of the crystal (in the sense of Kashiwara) in terms of nilpotent representations of the preprojective algebra.
  In the talk we present a construction to determine the irreducible components of the space of all locally nilpotent representations of the preprojective algebra using nilpotent classes. This space contains the space of nilpotent representations, and the irreducible components form a subset of the irreducible compenents of the space of all locally nilpotent representations.

Friday, 30 November 2012

• 13:15, Lecture Hall H6
  Valentin Katter (Bielefeld): Reduced representations in the representation ring of rooted tree quivers
  Abstract: For two representations V,W of a quiver Q we can define a pointwise tensor product. This tensor product together with the direct sum induces a ring structure on the set of isomorphism classes of representations of Q. We call this ring the representation ring of the quiver Q and denote it with R(Q). We can construct orthogonal idempotents and give a decomposition of R(Q) via the Möbius algebra on the partial ordered set of subquivers of Q. In this talk we will look at the ring structure of R(Q) for rooted tree quivers, which are quivers that have exactly one sink and whose underlying graph is a tree. Kinser discovered that for a rooted tree quiver, R(Q) modulo its nilpotent elements is a finitely generated Z-module, where the generators can be obtained by so called reduced representations. These reduced representations arise from a combinatorial construction and can be defined via the property that V is a direct summand of V².
• 14:30, Lecture Hall H6
  Greg Stevenson (Bielefeld): Gorenstein small categories and representations of finite projective dimension
  Abstract: We will discuss certain conditions on a small category C which ensure the category of representations of C, over a Gorenstein ring, is Gorenstein. In special cases, for instance mesh categories of simply laced Dynkin quivers, we will then demonstrate that, over a regular ring, one can characterise the representations of finite projective dimension in terms of exactness conditions coming from the structure of C. Time permitting, the motivating problem of finding universal coefficient theorems for triangulated categories will also be discussed. This is ongoing joint work with Ivo Dell'Ambrogio and Jan Stovicek.
  
• 16:00, Lecture Hall H6
  Thorsten Weist (Wuppertal): On the recursive construction of indecomposable quiver representations
  Abstract: Besides known techniques we investigate new techniques which can be used to construct indecomposable quiver representations recursively. These recursions come always along with a certain decomposition of some fixed root into smaller Schur roots. Often there exists a „well-behaved“ decomposition saying how to construct indecomposable representations. But one can also easily produce examples where it seems that only more complicated decompositions exist. This construction can also be used to construct indecomposable tree modules.

Friday, 23 November 2012

• 13:15, Lecture Hall H6
  Zhe Han (Bielefeld): The homotopy categories of injective modules of derived discrete algebras
  Abstract: The derived classification of algebras with discrete derived categories (derived discrete algebras) was given by Dieter Vossieck. Concerning the unbounded derived category D(Mod A) and the homotopy category K(Inj A) for some finite dimensional algebra A, I will give a characterization of generically trivial derived categories and a classification of the indecomposable objects in K(Inj A) for radical square zero algebras A which are derived discrete. As a consequence, all the indecomposable objects are endofinite.
  
• 14:30, Lecture Hall H6
  Henning Krause (Bielefeld): Discrete derived categories and Krull-Gabriel dimension
  Abstract: Discrete derived categories were introduced by Vossieck. In my talk I'll explain a conjecture, which says that a derived category is discrete iff its Krull-Gabriel dimension is bounded by one.
• 16:00, Lecture Hall H6
  Dieter Vossieck (Bielefeld): Representation-discrete algebras and the second Brauer-Thrall conjecture
  Abstract: We want to discuss the following statement which is equivalent to the second Brauer-Thrall conjecture / theorem of Nazarova-Roiter / theorem of Bautista-Bongartz:
  Over an algebraically closed field k , let k[[Q]] be the complete path-algebra of a finite quiver Q and let I be a closed ideal of k[[Q]] consisting of (possibly infinite) linear combinations of paths of length at least 2. Assume that k[[Q]]/I is minimal representation-infinite but admits only finitely many indecomposables of any given dimension. Then Q is an oriented cycle and I = 0 .

Friday, 09 November 2012

• 14:00, Lecture Hall H6
  Andreas Nickel (Bielefeld): Non-commutative Fitting invariants
  Abstract: One can associate to each finitely presented module M over a commutative ring R an ideal Fitt(M) which is called the (zeroth) Fitting ideal of M over R and which is an important natural invariant of M. For instance, it is always contained in the annihilator of M. We generalize this notion to orders over complete commutative noetherian local domains in separable algebras.

Friday, 02 November 2012

• Opening Bielefeld Graduate School in Theoretical Sciences
  16:00, Lecture Hall H2
  Christoph Schweigert (Hamburg): Large groups in mathematics, small scales in physics and a reason for finding integer numbers

Friday, 26 October 2012

• 14:00, Lecture Hall H6
  Sira Gratz (Hannover): Cluster Algebras of Infinite Rank
  Abstract: The combinatorics of a cluster algebra of type Q, where Q is an orientation of the Dynkin diagram A_n, can be expressed via triangulations of the (n + 3)-gon. As has been observed by Fomin and Zelevinsky, it follows that there is a cluster algebra structure of type Q on the homogeneous coordinate ring C[Gr(2, n + 3)] of the Grassmannian of planes, which is defined as the coordinate of the affine cone of Gr(2, n + 3) via the Plücker embedding. By allowing infinite countable clusters, this idea can be extended to the infinite case, motivated by results by Holm and Jørgensen, who have analysed a category, whose cluster tilting subcategories correspond to triangulations of the ∞-gon. We study the cluster algebra structures arising from the cluster structure on this category, obtaining infinite cluster algebra structures on the homogeneous coordinate ring C[Gr(2, ±∞)], where Gr(2, ±∞) is the space of planes in the pro-finite dimensional vectorspace k[[t,t−1]]. Moreover, the results of Grabowski and Launois on the quantum algebra structure on the quantum Grassmannian C_q[Gr(2,n)] can be generalized to the infinite case, yielding infinite quantum cluster algebra structures on C_q[Gr(2,±∞)].
• 15:15, Lecture Hall H6
  Jorge Vitória (Stuttgart): Ring epimorphisms and universal localisations
  Abstract: Ring epimorphisms are relevant to study certain subcategories of a fixed category of modules or of its derived category. One way to construct ring epimorphisms is to consider universal localisations, as defined by Cohn and Schofield. In this talk we will show that, in some cases, ring epimorphisms with a particularly nice homological property (so called homological ring epimorphisms) are precisely those given by universal localisations. Moreover, we will present a generalisation of universal localisation, introduced by Krause, and discuss necessary and sufficient conditions for its existence.

Friday, 19 October 2012

• 13:15, Lecture Hall H6
  Xiaojin Zhang (Nanjing): The Gorenstein projective conjecture
  Abstract: In this talk, we report that the Gorenstein projective conjecture is left and right symmetric and the co-homology vanishing condition can not be reduced in general. Moreover, the Gorenstein projective conjecture is true for CM-finite algebras.
• 14:30, Lecture Hall H6
  Luke Wolcott (Lisbon): Not every object in the derived category of a ring is Bousfield equivalent to a module
  Abstract: Given W and X in a tensor triangulated category, we say W is X-acyclic if W tensors with X to zero. Two objects X and Y are called Bousfield equivalent if they have the same acyclics. In this talk we give a (non-constructive) proof that there exist objects in the derived category of graded modules over a certain graded non-Noetherian ring that are not Bousfield equivalent to any module. This contrasts with the Noetherian case, and has consequences for subcategory classification.
  
• 16:00, Lecture Hall H6
  Giovanni Cerulli Irelli (Bonn): Desingularization of quiver Grassmannians
  Abstract: Given a Dynkin quiver Q and a (finite-dimensional) Q-representation M, let us consider a quiver Grassmannian X=Gr_e(M) associated with M, i.e. the projective variety of all subrepresentations of M of dimension vector e. This projective variety is not smooth in general, and its geometry is quite complicated. Our aim is to construct an explicit desingularization of X, i.e. a proper birational morphism f:\hat{X}->X from a smooth projective variety \hat{X}. The variety \hat{X} turns out to be again a quiver Grassmannian for a representation \hat{M} of an algebra $B_Q$ derived-equivalent to the Auslander algebra of kQ.
  
  This is a joint work with Evgeny Feigin and Markus Reineke (arXiv:1209.3960).
  

Friday, 12 October 2012

• 14:00, Lecture Hall H6
  Takuma Aihara (Bielefeld): On upper bounds of derived dimensions
  Abstract: The notion of the dimension of a triangulated category has been introduced by Rouquier. It measures how many extensions are needed to build the triangulated category out of a single object, up to finite direct sum, direct summand and shift. It is still a hard problem in general to give a precise value of the dimension of a given triangulated category. In the talk, we will focus on dimensions of bounded derived categories (derived dimensions) and give several upper bounds of derived dimensions.
• 15:15, Lecture Hall H6
  Daniel Labardini-Fragoso (Bonn): On a family of species with potentials
  Abstract: I will talk on work in progress, joint with Andrei Zelevinsky, regarding possible extensions of Derksen-Weyman-Zelevinsky's mutation theory of quivers with potentials to the setting where the matrix encoded by the quiver is not skew-symmetric but rather skew-symmetrizable.

Friday, 14 September 2012

• 14:00, Lecture Hall H11
  Thomas Hüttemann (Belfast): On finite domination of chain complexes
  Abstract: A chain complex of modules is called finitely dominated over the ring R if it is chain homotopy equivalent to a bounded complex of finitely generated projective R-modules. Finite domination has been considered extensively in algebra (finiteness conditions for groups) and topology (ends of manifolds).
  I will specifically consider the case of a bounded complex of finitely generated free modules over a Laurent polynomial ring R[x, 1/x, y, 1/y] in two indeterminates, and explain a homological criterion to decide whether the complex is finitely dominated over R. This joint project with David Quinn generalises one-variable results obtained by Andrew Ranicki, and makes use of algebraic data encoded by the geometry and combinatorics of a square. Time permitting I will indicate how n-variable versions can be attacked using more complicated combinatorics of higher-dimensional polytopes, and how our results are related to Sigma invariants in the sense of Bieri, Neumann and Strebel.
• 15:15, Lecture Hall H11
  Mohamed Barakat (Kaiserslautern): Computational algebraic geometry. From category theory to Gröbner basis and combinatorics
  Abstract: I will present an approach to constructive homological algebra based on the notion of a computable Abelian category. The algorithmic setup is general enough to cover applications ranging from linear control theory to algebraic geometry. It turns out that a big class of Serre quotient categories of coherent sheaves in toric algebraic geometry are computable in a strong sense. I will show how our approach connects abstract category theory to explicit Gröbner basis computations and combinatorics. My talk is based on recent joint work with Markus Lange-Hegermann
• 16:45, Lecture Hall H11
  Dirk Kussin (Verona): Large tilting sheaves over tubular exceptional curves
  Abstract: Let X be an exceptional curve. We consider the category Qcoh(X) of quasicoherent sheaves over X. A quasicoherent sheaf T is called tilting if Gen(T)=Ker Ext^1 (T,-). Tilting sheaves with coinciding Gen-classes are called equivalent. A tilting sheaf is called large, if T is not equivalent to a coherent tilting sheaf.
  We show that each tilting sheaf has a pure-injective torsion part. We classify all large tilting sheaves which have a non-coherent torsion part. In the tubular case we show that for every real number (and infinity) w there are large tilting sheaves of slope w, and we give a classification of them. This is joint work with Lidia Angeleri-Hügel.

Tuesday, 28 August 2012

• 14:00, Lecture Hall H11
  Ryo Kanda (Nagoya): Classifying Serre subcategories via atom spectrum
  Abstract: In this talk, we introduce the atom spectrum of an abelian category as a topological space consisting of all the equivalence classes of monoform objects. For an arbitrary commutative ring, we will see that the atom spectrum of the category of modules coincides with the prime spectrum as a set. In terms of the atom spectrum, we give a classification of Serre subcategories of an arbitrary noetherian abelian category.
  
• 15:15, Lecture Hall H11
  Leonid Positselski (Moscow): Contramodules and contraherent cosheaves
  Abstract: A contramodule is an object dual-analogous to a comodule,
  or to a torsion module, or to a discrete or a smooth module, or to
  a representation from the category O. A contraherent cosheaf is an object dual-analogous to a quasi-coherent sheaf. I will explain the definitions of a comodule and a contramodule over a coring over an associative ring, and give a couple of examples of representation-theoretic flavor. Time permitting, I will also explain what contramodules over topological rings are. Then I will proceed to the construction of the coring associated with an affine covering of an algebraic variety and the definition of contraherent cosheaves.
  
• 16:45, Lecture Hall H11
  Bin Zhu (Beijing): Cotorsion pairs in 2-Calabi-Yau triangulated categories
  Abstract: For a Calabi-Yau triangulated category of Calabi-Yau dimension d, the decomposition of this category as triangulated subcategories is determined by the decomposition of any d-cluster tilting subcategory. This induces that the quivers of cluster tilting objects in a 2-Calabi-Yau triangulated category are connected or are not connected at the same time. As an application, the t-structures (or co-t-structures) of an indecomposable 2-Calabi-Yau triangulated category with a cluster tilting object are proved to be trivial. This allows to give a classification of cotorsion pairs in this triangulated category.

Friday, 13 July 2012

• 14:15, Lecture Hall H10
  Markus Szymik (Bochum/Düsseldorf): The chromatic filtration of the Burnside category
  Abstract: In this talk I will discuss some aspects of the representation theory of finite groups from the homotopical point of view. The starting point is the Segal map which connects the Burnside category of finite groups to the stable homotopy category of their classifying spaces. I will focus on the interaction with the chromatic filtration of the latter. After first giving some history and background, I will then prove a related conjecture of Ravenel's in some cases, and present counterexamples to the general statement.

Friday, 06 July 2012

• NWDR Workshop
  10:30, Lecture Hall H16
  Andrew Hubery (Leeds): Braid group actions and presentations of affine quantum groups
  Abstract: We investigate the extent to which the affine quantum group is determined by the two properties of containing the corresponding quantum group of finite type and having an action of the (extended) affine braid group. Using a presentation of the braid group (analogous to the presentation of the extended affine Weyl group as the semi-direct product of the finite Weyl group and the weight lattice), we see that this is well adapted to Drinfeld's new presentation of the quantum group and thus obtain a conceptual proof of the isomorphism between this and the Drinfeld-Jimbo presentation.
• NWDR Workshop
  11:45, Lecture Hall H16
  Antoine Touzé (Paris): Finite generation of the cohomology of reductive group schemes
  Abstract: It has been proved by Haboush that if G is a reductive group scheme acting on an algebra A commutative and finitely generated, then the invariants A^G form a finitely generated algebra (this was conjectured by Mumford).
  Evens (1964), and later Friedlander Suslin (1997), proved similar finite generation theorems for the cohomology ring H^*(G,A) for finite groups and finite group schemes G acting on a commutative finitely generated algebra A. After this, van der Kallen conjectured that all the reductive group schemes have finitely generated cohomology algebras.
  In this talk we will present an overview of the proof of this theorem and how strict polynomial functors come into play for this problem.
  NB: The finite generation result depends (directly or indirectly) on the contribution of many authors: Friedlander, Suslin, van der Kallen, Srinivas, Grosshans and Touzé, and a complete proof is available in the article:
  "Bifunctor cohomology and Cohomological finite generation for reductive groups" Duke Math. J.  151  (2010).
• NWDR Workshop
  14:00, Lecture Hall H10
  Ulrich Krähmer (Glasgow): Batalin-Vilkovisky Structures on Ext and Tor
  Abstract: The topic of this talk (based on joint work with Niels Kowalzig) is an algebraic structure whose best known example is provided by the multivector fields and the differential forms on a smooth manifold: the multivector fields are a Gerstenhaber algebra with respect to wedge product and Schouten-Nijenhuis bracket, and the differential forms are what we call a Batalin-Vilkovisky module over this Gerstenhaber algebra, which means that the multivector fields act in two ways on forms - by means of contraction and of Lie derivative - and that these actions are related by a differential that fits into Cartan's "magic" homotopy formula. Nest, Tamarkin and Tsygan suggested to refer to this abstract package of two graded vector spaces with such operations as to a noncommutative differential calculus.
  Generalising work by the above mentioned authors and by Getzler, Gerstenhaber, Goodwillie, Huebschmann, Rinehart and others, I will explain that Ext and Tor over Hopf algebroids tends to carry such a structure which means that homological algebra produces plenty of examples of noncommutative differential calculi, including for example Hochschild and Poisson (co)homology.
  As a first application, Ginzburg's theorem that the Hochschild cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra is extended to twisted Calabi-Yau algebras such as quanum groups, quantum homogeneous spaces or quantum vector spaces.
• NWDR Workshop
  15:30, Lecture Hall H10
  Hanno Becker (Bonn): Models for Singularity Categories
  Abstract: This talk is about the construction of various Quillen models for categories of singularities. I will begin by recalling the connection between abelian model structures, cotorsion pairs and deconstructible classes and then describe the construction of the singular models. Next, I will explain how Krause's recollement for the stable derived category can be obtained model categorically. Finally, as an example I will show that Positselski's contraderived model for the homotopy category of matrix factorizations is Quillen equivalent to a particular singular model structure on the category of curved mixed complexes.
• NWDR Workshop
  16:45, Lecture Hall H10
  Bill Crawley-Boevey (Leeds): D-modules for nodal curves and multiplicative preprojective algebras
  Abstract: In work on the Deligne-Simpson problem, I introduced (with Shaw), certain algebras, called multiplicative preprojective algebras, and I also studied monodromy of logarithmic connections on vector bundlles on the Riemann sphere.
  In order to directly connect these notions, I showed in arXiv:1109.2018 [math.RA], that multiplicative preprojective algebras provide a natural receptacle for monodromy for certain systems of vector bundles, linear maps and logarithmic connections on what I called 'Riemann surface quivers'.
  Instead of this perhaps artificial notion, I was asked whether or not it could be formulated in terms of torsion-free sheaves on nodal curves. This talk is a response to that question. I will discuss various categories of D-modules on nodal curves, and then a modification which is related to multiplicative preprojective algebras.

Friday, 29 June 2012

• 14:30, Lecture Hall H10
  Dong Yang (Stuttgart): The heart of a t-structure: some examples
  Abstract: Let A be a finite-dimensional algebra. Then there is a one-to -one correspondence between silting objects in K^b(proj A) and bounded t-structures on D^b(A) with length heart. I will talk about some applications of this result.
• 16:00, Lecture Hall H10
  Christian Stump (Hannover): Revisiting the combinatorics of cluster categories in finite types
  Abstract: I will present the natural combinatorial construction of subword complexes and describe their connections to repetition quivers, Auslander-Reiten quivers, and cluster categories in finite types. This perspective leads to new combinatorial objects called multi-cluster complexes, of which I will discuss possible relations to certain identifications in the repetition quiver in terms of Auslander-Reiten translates and shifts.

Friday, 22 June 2012

• 13:15, Lecture Hall H10
  Reiner Hermann (Bielefeld): An adaptation of Schwede's loop bracket construction for monoidal categories
  Abstract: In the late 90's, Stefan Schwede established a categorial interpretation of the Lie structure in Hochschild cohomology using some bilinear operation which he called the "loop bracket". In this talk, we will explain how to imitate this construction for exact categories admitting a (semi-exact) monoidal structure. For this purpose, we will introduce mutually inverse isomorphisms relating the zeroth and the first homotopy groups of extension categories. This partially recaptures a result by Retakh published in a 1986 paper. It will turn out, that, provided the underlying monoidal category admits a braiding, the loop bracket will be trivial, leading to interesting insights relating to, for example, the Lie structure of the cohomology ring of a cocommutative Hopf algebra.
• 14:30, Lecture Hall H10
  Yong Jiang (Bielefeld): Every projective variety is a quiver Grassmannian (after Reineke)
  Abstract: This is a report on a short paper of Reineke. We will show that every projective variety can be realized as a quiver Grassmannnian, for an acyclic quiver Q with at most three vertices, a Schurian representation and a particular dimension vector.
  
• 16:00, Lecture Hall H10
  Martin Kalck (Bonn): (Relative) Singularity categories
  Abstract: There are two interesting triangulated categories associated with any (MCM-representation finite) Gorenstein singularity: the singularity category of Buchweitz and the relative singularity category of a non-commutative (Auslander) resolution, which was studied in joint work with Burban. We show that these categories mutually determine each other in the case of ADE-singularities in any Krull dimension. Knörrer's periodicity theorem yields a wealth of non-trivial examples. This is joint work with Dong Yang.

Friday, 15 June 2012

• 13:15, Lecture Hall H10
  Raquel Simoes (Leeds): Hom-configurations and noncrossing partitions
  Abstract: We will give a classification of maximal Hom-free sets of indecomposable objects (the Hom-configurations in the title) in a certain orbit category of the bounded derived category of a path algebra of Dynkin type in terms of noncrossing partitions.This classification generalizes a result of C. Riedtmann arising in her work on representation-finite selfinjective algebras of tree class A.

Friday, 08 June 2012

• 14:30, Lecture Hall H10
  Johan Steen (Trondheim): Orlov's spectrum of triangulated categories
  Abstract: In a 2009 paper, Orlov introduced an invariant of triangulated categories he called the "dimension spectrum". Roughly speaking, an integer d belongs to this spectrum iff there is an object from which you can generate the whole category by taking cones precisely d times. This is a natural extension of the Rouquier-dimension which is the smallest such integer. In a recent paper by Ballard--Favero--Katzarkov, general methods for obtaining information about the dimension spectrum were obtained. We will introduce some of these methods and pay special attention to examples from representation theory.
• 16:00, Lecture Hall H10
  Shawn Baland (Aberdeen): The generic kernel filtration for modules of constant Jordan type
  Abstract: Let E be an elementary abelian p-group of rank two and k an algebraically closed field of characteristic p. Recently, Carlson, Friedlander and Suslin have constructed a canonically defined submodule of a kE-module called the generic kernel. In the case where M is a kE-module of constant Jordan type, they have shown that the generic kernel admits a filtration of M in which many of the terms have constant Jordan type. In this talk I will introduce a duality formula for subquotients in the above filtration and answer the authors' question regarding whether or not all filtration terms have constant Jordan type.

Friday, 04 May 2012

• 14:15, Lecture Hall H10
  Nils Mahrt (Bielefeld): Idempotents in Representation Rings of Quivers
  Abstract: This is a report on work of Herschend, Kinser and Schiffler. Let Q be a quiver and k an algebraically closed field. We will define the representation ring R(Q) as follows: On the split Grothendieck group of the category of representations of Q define a multiplication induced by the pointwise tensor product of quiver representations. For an acyclic quiver Q we will construct certain orthogonal idempotent elements in R(Q).

Friday, 20 April 2012

• 14:15, Lecture Hall H10
  David Pauksztello (Hannover): Co-t-structures and co-stability
  Abstract: In this talk we introduce the ideas of co-t-structures and co stability conditions and compare and contrast with t-structures and stability conditions. We show that the space of co-stability conditions on a triangulated category forms a complex manifold, and give some examples.

Friday, 13 April 2012

• 14:15, Lecture Hall H10
  Claus Michael Ringel (Bielefeld): Morphisms determined by objects: The case of modules over artin algebras
  Abstract: Let R be an artin algebra. In his Philadelphia Notes, Auslander showed that any homomorphism between R-modules is right determined by an R-module C, but a formula for C which he wrote down has to be modified. The lecture will discuss the indecomposable direct summands of the minimal right determiner of a morphism, in paricular the role of the projective direct summands. We will provide a detailed analysis of those morphisms which are right determined by a module without any non-zero projective direct summand. What we encounter is an intimate relationship to the vanishing of Ext^2.

Friday, 23 March 2012

• 13:15, Room U2-205
  Jan Schröer (Bonn): Generic Caldero-Chapoton functions and generalized clusters
  Abstract: This is joint work with Giovanni Cerulli Irelli and Daniel Labardini-Fragoso. To any algebra A defined as a factor of a completed path algebra we associate an algebra CC(A) generated by generalized Caldero-Chapoton formulae. We construct a candidate G(A) for a basis of CC(A), and we show that G(A) is linearly independent. All elements in G(A) are generalized cluster monomials. Then we apply these results to the theory of cluster algebras. Even for acyclic cluster algebras one obtains a much richer structure than the classical cluster algebra theory can provide. We use the concept of strongly reduced components of module varieties introduced by Geiss, Leclerc and Schröer, and generalizations of recent results by Plamondon, who parametrizes strongly reduced components for finite-dimensional algebras.
• 14:30, Room U2-205
  Dave Benson (Aberdeen): Around a theorem of Mislin in the cohomology of finite groups
  Abstract: Mislin proved in 1990 that the inclusion of a subgroup H in a finite group G induces an isomorphism in mod p cohomology if and only if the index is prime to p and H controls fusion in G. His proof was essentially topological in nature, as it used the then recent proof of the Sullivan conjecture as well as results of Dwyer and Zabrodsky. Peter Symonds partly algebraised the proof in 2004, and the rest of the proof was recently algebraised by Hida and Okuyama in terms of some rather intricate arguments with cohomology of trivial source modules.
  It turns out that if p is odd, a much stronger statement is true. Namely, if |H:G| is prime to p and the inclusion just induces an F-isomorphism in mod p cohomology (i.e., the kernel is nilpotent and every element has some p-power power in the image) then H controls fusion in G; and therefore the inclusion is actually a cohomology isomorphism. Ellen Henke recently provided the final fusion theoretic argument that completes the purely algebraic proof of this stronger statement.
• 16:00, Room U2-205
  Jon Carlson (Athens, Georgia): Thick subcategories of the bounded derived category
  Abstract: This is joint work with Srikanth Iyengar. It is all about using methods from commutative algebra to study group representations. A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike the previous proofs of these results.

Friday, 10 February 2012

• 14:30, Room U2-205
  Jan Stovicek (Prague): Resolving subcategories for a commutative noetherian ring
  Abstract: Given a commutative notherian ring R, we classify resolving subcategories of mod-R consisting of modules of bounded projective dimension in terms of certain descending sequences of specialization closed subsets of the Zariski spectrum of R. This is a consequence of a similar classification result for cotilting classes in the category of infinitely generated modules over R. The talk is an account on joint work with L. Angeleri, D. Pospisil and J. Trlifaj.
• 16:00, Room U2-205
  Markus Perling (Bochum): Equivariant resolutions and maximal Cohen-Macaulay modules over affine toric varieties
  Abstract: We present a combinatorial framework which allows to translate free resolutions of Z^n-graded modules over the polynomial ring in n variables into projective resolutions over certain incidence algebras. We present two applications: 1. we explicitly determine the Betti-numbers and local cohomologies of certain such modules related to hyperplane arrangements; 2. we produce new examples of MCM modules over certain toric rings.

Friday, 27 January 2012

• 14:30, Room V2-213
  Hongmei Zhao (Nankai): On the structure of the augmentation quotient groups for some nonabelian groups
  Abstract: Let G be a finite group, ZG its integral group ring and delta^n(G) the n'th power of the augmentation ideal delta(G), denote Q_n(G)=delta^n(G)/delta^{n+1}(G) the augmentation quotient groups of G. We consider the dihedral group D_{2^tk}(k odd) and m'th symmetric group S_m, we show Q_n(D_{2^tk}) is an elementary 2-group and its rank is no more than 2t+1. As for Q_n(S_m), we have Q_n(S_m) is isomorphic with Z_2.
• 16:00, Room V2-213
  Barbara Baumeister (Bielefeld): Some aspects of group theory 200 years after Galois
  Abstract: Galois introduced the notion of a group to solve old geometric questions. In the talk I will show that this concept is still alive by discussing recent results on twin trees as well as on permutation polytopes.

Friday, 20 January 2012

• 13:15, Room V2-213
  Shoham Shamir (Bergen): A colocalization spectral sequence
  Abstract: Colocalization is a right adjoint to the inclusion of some subcategory. Given a differential graded algebra R, it is natural to ask for a spectral sequence which connects a colocalization in the derived category of R-modules and an appropriate colocalization in the derived category of graded modules over the cohomology ring of R. It turns out that, under suitable conditions, such a spectral sequence exists. This generalizes the Greenlees spectral sequence. I will describe this generalization and show some applications.
• 14:30, Room V2-213
  Irakli Patchkoria (Bonn): On the algebraic classification of module spectra
  Abstract: For any S-algebra R whose homotopy ring is sufficiently sparse and has graded global homological dimension less or equal than three, we construct an equivalence between the derived category of R and the derived category of its homotopy ring. This improves Bousfield-Wolbert algebraic classification of isomorphism classes of objects in the derived category of R. In the case of global dimension two, the p-local real connective K-theory, the first Johnson Wilson spectrum E(2) and the truncated Brown-Peterson spectrum BP<1>, for an odd prime p, serve as our main examples. Examples of S-algebras with three dimensional homotopy ring to which our result applies are E(3) and BP<2> at a prime greater or equal than five.

Friday, 16 December 2011

• 14:30, Room V2-213
  Claus Michael Ringel (Bielefeld): Representations of a quiver over the algebra of dual numbers
  Abstract: The representations of a quiver Q over a field k have been studied for a long time, and one knows quite well the structure of the category of kQ-modules. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. The lecture will draw the attention to the case when A = k[epsilon] is the algebra of dual numbers, thus to the category of Lambda-modules, where Lambda = kQ[epsilon] is the path algebra of Q over A. The algebra Lambda is a 1-Gorenstein algebra, thus the torsionless Lambda-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen-Macaulay modules). They form a Frobenius category L, thus the corresponding stable category is a triangulated category T. This category T is triangle equivalent to the orbit category of the bounded derived category of the kQ-modules modulo the shift. The homology functor H yields a bijection between the indecomposables in T and those in mod kQ, the inverse is given by forming the minimal L-approximation. We also describe the embedding of the Auslander-Reiten quiver of mod kQ into that of T.
  
  This is a report on current joint investigations with Zhang Pu (SJTU).
• 16:00, Room V2-213
  Gena Puninski (Moskow): The Ziegler spectrum of domestic string algebras (Part II)
  Abstract: The Ziegler spectrum of a ring is a topological space whose points are indecomposable pure injective modules and basic open sets are determined by morphisms between finitely presented modules. This space captures a lot of information on the category of modules and there are many applications of this notion.
  In this series of talks we will describe some general method of classifying indecomposable p.i. modules and apply it to describe the Ziegler spectrum of string algebras. In the case of 1-domestic string algebra we will completely classify the points and the topology of Ziegler spectrum, confirming in particular a conjecture made by Ringel. We will also hope to prove that the Krull-Gabriel dimension of any 1-domestic string algebra does not exceed 3 (Schroer's conjecture).
  We will discuss a general approach to classifying arbitrary (not necessarily indecomposable) pure injective modules over string algebras, most prominent of those are superdecomposable modules, and formulate some global conjectures on their structure.
  The lectures will be given in `proof by example' style, with many diagrams drawn; no previous knowledge of the Ziegler spectrum is expected, though some acquaintance with representation theory will help.

Friday, 02 December 2011

• 13:15, Room V2-213
  Artem Lopatin (Omsk): Matrix identities with forms
  Abstract: A linear group GL(n) acts on d-tuples of n x n matrices by simultaneous conjugation. The algebra R(n,d) of polynomial invariants of this action is called the algebra of matrix GL(n)-invariants. In the case of arbitrary characteristic of the base field Donkin [Invent. Math. 110 (1992), 389-401] established generators of R(n,d) and Zubkov [Algebra and Logic 35 (1996), No. 4, 241-254] described relations between them. Namely, Zubkov showed that the ideal of relations is generated by the coefficients $\sigma_k$ of the characteristic polynomial of a matrix for k>n. We proved that the ideal of relations is actually generated by $\sigma_k$ for $n<k\leq 2n$. In particular, we showed that the T-ideal of identities of $M_n$ with forms is finitely based.
• 14:30, Room V2-213
  Grzegorz Bobinski (Torun): Semi-invariants of concealed-canonical algebras
  Abstract: The descriptions of the algebras of semi-invariants for regular dimension vectors over Euclidean and canonical algebras are basically identical. Thus it is natural to expect that these descriptions generalize to arbitrary concealed-canonical algebras. The main result of my talk states that this is really the case.
• 16:00, Room V2-213
  Gena Puninski (Moskow): The Ziegler spectrum of domestic string algebras (Part I)
  Abstract: The Ziegler spectrum of a ring is a topological space whose points are indecomposable pure injective modules and basic open sets are determined by morphisms between finitely presented modules. This space captures a lot of information on the category of modules and there are many applications of this notion.
  In this series of talks we will describe some general method of classifying indecomposable p.i. modules and apply it to describe the Ziegler spectrum of string algebras. In the case of 1-domestic string algebra we will completely classify the points and the topology of Ziegler spectrum, confirming in particular a conjecture made by Ringel. We will also hope to prove that the Krull-Gabriel dimension of any 1-domestic string algebra does not exceed 3 (Schroer's conjecture).
  We will discuss a general approach to classifying arbitrary (not necessarily indecomposable) pure injective modules over string algebras, most prominent of those are superdecomposable modules, and formulate some global conjectures on their structure.
  The lectures will be given in `proof by example' style, with many diagrams drawn; no previous knowledge of the Ziegler spectrum is expected, though some acquaintance with representation theory will help.

Friday, 25 November 2011

• 16:00, Room C0-269
  Christopher Voll (Bielefeld): Representation zeta functions of arithmetic groups
  Abstract: A group is called (representation) rigid if it has, for each n, only finitely many irreducible complex representations of dimension n. To study the representation growth of rigid groups is to study the arithmetic and asymptotic properties of the number of such representations, as n tends to infinity. If these numbers grow at most polynomially, a profitable approach to their study is to encode them in a Dirichlet generating series - the group's representation zeta function. Under additional assumptions, such zeta functions have Euler products indexed by places in algebraic number fields. The factors of such Euler products can be studied using a wealth of methods from geometry and combinatorics. Major questions regarding representation zeta functions of groups ask about properties of the Euler factors, such as rationality, and local and global abscissae of convergence.
  
  I will report on recent joint work on representation zeta functions of arithmetic groups. In a project with Avni, Klopsch and Onn, we establish a conjecture of Larsen and Lubotzky on the abscissae of convergence of irreducible lattices in higher-rank semisimple groups. In joint work with Stasinski we give uniform formulae for representation zeta functions of finitely generated nilpotent groups. A common feature of both projects is the use of sophisticated machinery from the theory of p-adic integration and a Kirillov orbit method to parameterize representations by co-adjoint orbits.

Friday, 18 November 2011

• 13:15, Room V2-213
  Sarah Scherotzke (Bonn): The Integral Cluster Category
  Abstract: In my talk, we will consider the question when orbit categories of triangulated categories are again triangulated. I will present some examples where this fails and give a sufficient condition proven by Bernhard Keller for the orbit category of a triangulated category to have a natural triangulated structure. Applying this result to the Cluster category associated to a finite acyclic quiver over a field shows that it is triangulated. In joint work with Bernhard Keller, we proved that the Cluster category defined over certain commutative rings are triangulated, we classify the Cluster-tilting objects and show that they are linked by mutation. The proof in the integral case does not use Keller's criteria and requires a different approach of which I will give sketch.
• 14:30, Room V2-213
  Roger Wiegand (Lincoln): Brauer-Thrall theorems and conjectures for commutative local rings
  Abstract: The Brauer-Thrall Conjectures, now theorems, were originally formulated in terms of representations of finite-dimensional algebras. They say, roughly speaking, that failure of finite representation type entails the existence of lots of indecomposable representations of large dimension. These conjectures have been successfully transplanted to the representation theory of commutative local rings. This talk will be a survey of such results, conjectures and counterexamples, for various categories of finitely generated modules over a commutative Noetherian local ring. The emphasis will be on maximal Cohen-Macaulay modules over Cohen-Macaulay local rings.
• 16:00, Room V2-213
  Sylvia Wiegand (Lincoln): The Anatomy of a Stranger: A (Barely) non-Noetherian Ring
  Abstract: In ongoing work with William Heinzer and Christel Rotthaus over the past twenty years we have been applying a construction technique for obtaining sometimes baffling, sometimes badly behaved, sometimes Noetherian, sometimes non-Noetherian integral domains. This technique of intersecting fields with power series rings goes back to Akizuki in the 1930s and Nagata in the 1950s; since then it has been employed by Nishimuri, Heitmann, Ogoma, the authors and others.
  In particular we show how to obtain a three-dimensional near-Noetherian unique factorization domain B that is tantalizingly close to being Noetherian but is not quite, because exactly one prime ideal has height two and it is the only nonfinitely generated prime ideal of B. The unique maximal ideal of B is 2-generated. We also mention more mysterious generalizations to higher dimensions.

Friday, 04 November 2011

• 14:30, Room V2-213
  Yong Jiang (Bielefeld): Parametrizations of canonical bases and irreducible components of nilpotent varieties
  Abstract: It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis. For each reduced expression of an element in the Weyl group, Geiss, Leclerc and Schroeer have recently given a parametrization of the set of irreducible components in studying the cluster structure of the coordinate ring of the corresponding unipotent subgroup. We show that their parametrization is compatible with Lusztig's parametrization of canonical basis. And we also give some interpretations of Lusztig's transition maps.
  
• 16:00, Room V2-213
  Moritz Groth (Nijmegen): On the theory of derivators
  Abstract: The theory of derivators -- going back t Grothendieck and Heller -- is a purely (2-)categorical approach to an axiomatic homotopy theory. The usual passage from a model category (resp. an abelian category) to the underlying homotopy category (resp. derived category) result in a loss of information. The typical defects of triangulated categories (e.g. the non-functoriality of the cone construction) can be seen as a reminiscent of this fact. The basic idea of a derivator is that one should instead simultaneously form homotopy/derived categories of `all' diagra categories and also keep track of the restriction and homotopy Kan extensio functors. The aim of this talk is to give an introduction to the theory of derivators and t (hopefully) advertise it as a convenient 'weakl terminal' approach to axiomatic homotopy theory.
  

Friday, 28 October 2011

• 14:30, Room V2-213
  Anna-Louise Paasch (Bielefeld): Monoid algebras of projection functors
  Abstract: For each simple representation of a finite acyclic quiver there is a projection functor onto the kernel of the covariant Hom-functor of the simple. We study the monoid (algebra) generated by those projection functors w.r.t. composition.
  The monoid algebra associated with a linearly oriented Dynkin quiver of type A is discussed in detail. We determine defining relations for several other cases and illustrate the influence of the orientation rather than the representation type of the underlying graph.
• 16:00, Room V2-213
  Ralf Meyer (Göttingen): Hereditary exact categories from equivariant bivariant K-theory
  Abstract: Together with Ryszard Nest and my students Rasmus Bentmann and Manuel Köhler, I have been studying Universal Coefficient Theorems in bivariant K-theory for C*-algebras. The idea there is to find a K-theoretic invariant that completely classifies certain diagrams of C*-algebras up to weak equivalence. With a guess for this invariant in hand, a general machinery for homological algebra in triangulated categories provides the required Universal Coefficient Theorem provided certain modules over a certain ring have projective resolutions of length 1. As a result, we proved several positive and negative results about the existence of such resolutions for certain rings.
  The results we obtained hint at a connection with quiver representations, in particular, the special features of ADE-quivers, but we do not yet understand this. I hope that discussions with algebraists can clarify this relationship and what kind of results to expect in more general cases.

Friday, 21 October 2011

• 13:15, Room V2-213
  Jesse Burke (Bielefeld): Finite injective dimension over rings with Noetherian cohomology
  Abstract: After discussing rings with Noetherian cohomology, examples of which include group rings of finite groups and complete intersection rings, we will state and prove a criterion for a complex over such a ring to have finite injective dimension. The criterion generalizes a theorem in the representation theory of finite groups and a theorem of Avramov-Buchweitz for complete intersection rings. The proof uses the support theory of Benson, Iyengar, and Krause.
  
• 14:30, Room V2-213
  Philipp Lampe (Bielefeld): Cluster theory and Lusztig's canonical basis
  Abstract: With a Dynkin quiver one associates two triangulated categories: the cluster category and stable category of the module category of the preprojective algebra. We will explain why the categories provide an additive categorification of cluster algebras. They are related to various bases of the universal enveloping algebra of the same type. The talk concerns especially the connection between cluster variables and the dual canonical basis.
• 16:00, Room V2-213
  Robert Marsh (Leeds): Cluster presentations of reflection groups (joint work with Michael Barot, UNAM, Mexico)
  Abstract: We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type.

Friday, 07 October 2011

• 14:15, Room V4-119
  Gil Kaplan (Tel Aviv): Some new results on finite T-groups
  Abstract: A group is called a T-group if all its subnormal subgroups are normal, or, equivalently, if normality is a transitive relation among its subgroups. The subject was extensively studied since the seminal paper by Gaschuetz (1957), in which he described the structure of finite solvable T-groups. We give new results in this subject and new characterizations of solvable T-groups. One characterization involves properties of maximal subgroups. A second characterization involves product of conjugate subgroups.

Friday, 16 September 2011

• 14:15, Lecture Hall H10
  Dave Benson (Aberdeen): Modules of constant Jordan type and a conjecture of Rickard
  Abstract: I shall introduce modules of constant Jordan type for elementary abelian p-groups, and explain various conjectures about the possible Jordan canonical forms for such modules. One such conjecture was formulated by Jeremy Rickard in 2008. I shall outline the proof of a special case of this conjecture, namely that if a module of constant Jordan type has no Jordan blocks of length one, then the total number of Jordan blocks is divisible by p. I shall also outline some consequences of this conjecture.

Friday, 02 September 2011

• 14:00, Lecture Hall H10
  Dirk Kussin (Verona): On localization, tilting and torsion
  Abstract: Torsion theories and (co-) tilting objects in Mod-A for an hereditary algebra are described with the notion of universal localizations. In this talk an alternative approach via Qcoh-X and Ore localizations in noncommutative graded rings is discussed. This is about joint work with Lidia Angeleri.
  
• 15:15, Lecture Hall H10
  Phillip Linke (Bielefeld): Computational approach to the Artinian conjecture
  Abstract: First part of my talk will be dedicated to an introduction to generic representation theory. I will describe some of the concepts used and also introduce the mentioned conjecture. Afterwards I will explain how I used computational methods to determine some of the structure of the underlying category and how this can be used to approach the overall goal. In the end I will talk about the Gabriel quiver of a category as a nice application of some of the work that was done before.
  
• 16:45, Lecture Hall H10
  Hideto Asashiba (Shizuoka): A characterization of derived equivalences of oplax functors and string diagrams
  Abstract: We fix a small category I and a commutative ring k , and denote by k-Cat the 2-category of k-categories. An (op)lax functor from I to k-Cat can be regarded as a generalization of a lax group action on a k-category. We explain those oplax functors and derived equivalences of them using string diagrams that have been used to describe motions of particles in physics, and a proof of our Morita type theorem characterizing derived equivalences of oplax functors.
• 18:00, Lecture Hall H10
  Serge Bouc (Amiens): The slice Burnside ring and the section Burnside ring of a finite group.
  Abstract: I will introduce two new Burnside rings for a finite group G, built as Grothendieck rings of categories of morphisms of G-sets, and Galois morphisms of G-sets, respectively, and show how the properties of the usual Burnside ring can be extended to these new rings.

Friday, 22 July 2011

• Seminar: Noncrossing partitions

Thursday, 21 July 2011

• 10:30, Lecture Hall H8
  Helene Tyler (Riverdale): Auslander-Reiten Layers in the Rhombic Picture
  Abstract: The Gabriel-Roiter measure was first introduced by Roiter in his 1968 proof of the first Brauer-Thrall conjecture. For a finite length module, the pair consisting of the GR measure and the GR-comeasure defines the position of the module in the rhombic picture, as defined by Ringel. I will present a "greedy algorithm" for constructing these measures for quivers of type $\tilde{\mathbb A}_n$ and show that for such quivers, the measure decomposes into three characteristic parts. For an arbitrary hereditary algebra, we will see that modules lying on the intersection of a ray and a coray in a stable tube in the Auslander Reiten quiver correspond to limit points in the rhombic picture. Moreover, we will see the connection between the positions of the limit points in the rhombic picture and the Auslander-Reiten sequences in the tubes. Illustrative examples will be presented via the greedy algorithm. This talk reflects joint work with Markus Schmidmeier.
• 13:00, Lecture Hall H8
  Julia Sauter (Leeds): An affine cell decomposition of A_n-equioriented quiver flag varieties
  Abstract: We study quiver flag varieties for the A_n-equioriented quiver in analogy to classical Springer fibres. For classical Springer fibres we have a stratification parametrized by standard tableau and a refinement to affine cell decompositions parametrized by row tableau. We can find analogue stratifications parametrized by certain (skew) tableau in the A_n-equioriented case which give us a basis of their cohomology groups.
• 14:15, Lecture Hall H8
  Sefi Ladkani (Bonn): Derived equivalence classification of the gentle algebras arising from surface triangulations
  Abstract: The gentle algebras arising from surface triangulations have been introduced by Assem, Bruestle, Charbonneau-Jodoin and Plamondon. They can be seen as finite-dimensional Jacobian algebras of quivers with potentials, where a quiver corresponding to an ideal triangulation of a surface with marked points on its boundary has been defined by Fomin, Shapiro and Thurston and a potential has been defined by Labardini-Fragoso.
  We will present a complete derived equivalence classification of these algebras, thereby generalizing the classifications of cluster-tilted algebras of Dynkin type A by Buan and Vatne and those of affine type A tilde by Bastian, which correspond to the cases where the surface is a disk or an annulus, respectively.
  A crucial role in the classification is played by a derived invariant introduced by Avella-Alaminos and Geiss for gentle algebras and computed recently for the algebras in question by David-Roesler and Schiffler on the one hand, and the theory of good mutations introduced by the speaker on the other hand.
  If time permits, we will also indicate connections to the combinatorial problem of determining the mutation classes of quivers with the property that all their members have the same number of arrows.
• 17:00, Lecture Hall H8
  Ragnar-Olaf Buchweitz (Toronto): Variations on a Theorem of Orlov, II
  Abstract: See July 20.

Wednesday, 20 July 2011

• Seminar: Noncrossing partitions
• 11:15, Lecture Hall H8
  Ragnar-Olaf Buchweitz (Toronto): Variations on a Theorem of Orlov, I
  Abstract: In these talks we will discuss the proof and scope of Orlov's theorem comparing various Verdier quotients of suitable triangulated categories of (multi-)graded modules, among them derived categories of coherent sheaves on noncommutative projective stacks and stable categories of maximal Cohen-Macaulay modules, equivariant for a suitable group action.
  We will explain how Grothendieck-Serre duality makes the original proof more transparent.
  Orlov's results give rise to various decompositions and new invariants for (complexes of) graded modules. Will discuss these, present some examples, and ask some questions that need further research.

Friday, 15 July 2011

• 13:15, Lecture Hall H10
  Ivo Dell'Ambrogio (Bielefeld): The derived category of a graded commutative noetherian ring, Part I
  Abstract: For any graded commutative noetherian ring, where the grading group is finitely generated abelian and where commutativity is allowed to hold in a quite general sense, we establish an inclusion-preserving bijection between, on the one hand, the twist-closed localizing subcategories of the derived category, and, on the other hand, subsets of the homogeneous spectrum of prime ideals of the ring.
  The goal of the talk is to explain precisely what this means and to give some details concerning the proof.
• 14:30, Lecture Hall H10
  Greg Stevenson (Bielefeld): The derived category of a graded commutative noetherian ring, Part II
  Abstract: I will finish giving details on the proof of the classification of tensor ideals in the derived category of graded modules over a graded commutative noetherian ring. Then we will discuss examples and applications of the main result. In particular I will explain how one can obtain easily from this theorem classification results for derived categories of certain projective schemes without needing to treat more general non-affine schemes.
  
• 16:00, Lecture Hall H10
  Mike Prest (Manchester): Superdecomposable pure-injectives over tubular algebras
  Abstract: A module over an artin algebra is pure-injective (also termed algebraically compact) iff it is a direct summand of a direct product of finite length modules. Existence of superdecomposable (i.e. without indecomposable summands) pure-injectives can be seen as an indication of complexity of a module category. Richard Harland, in his thesis, showed that if R is a tubular algebra then, for each irrational slope there is a superdecomposable pure-injective of that slope (at least if R is countable; in general the result can be said in terms of a certain dimension). I present this and related results.

Friday, 01 July 2011

• 13:15, Lecture Hall H10
  Grzegorz Bobinski (Torun): Normality of maximal orbit closures for Euclidean quivers
  Abstract: Given a quiver Q and a dimension vector d one studies singularities appearing in the orbit closures with respect to the action of the product GL(d) of general linear groups on the space of representations with dimension vector d. We have shown in a joint paper with Zwara that these closures are normal Cohen-Macaulay varieties if Q is of Dynkin type A/D (the case of type E is not settled yet). On the other hand, Zwara gave an example showing that there exist orbit closures which are neither normal nor Cohen-Macaulay for quivers of infinite type. His example is of very special form and in my talk I will present the result stating that the closures of maximal orbits are still normal and Cohen-Macaulay if Q is of Euclidean type.
  
• 14:45, Lecture Hall H10
  Dolors Herbera (Barcelona): The combinatorics around an infinite idempotent matrix
  Abstract: I want to explain some tools developed recently to classify countably generated projective modules over some classes of rings. The ideas are based on a clever way to understand infinite column-finite idempotent matrices.
  I will emphasize on the case of semilocal rings, that is, rings that are semisimple artinian modulo its Jacobson radical. The developed theory allows to understand projective modules over semilocal noetherian rings but the general case is still quite challenging.

Friday, 10 June 2011

• 13:15, Lecture Hall H4
  Matthias Warkentin (Chemnitz): "Categorifications" of quiver mutation
  Abstract: Quiver mutation provides the combinatorial basis for cluster algebras introduced by Fomin and Zelevinsky. Work of Buan, Marsh, Reiten, Hubery and others establishes rich connections with representation theory. In this talk we will explain some of these connections and a nice combinatorial application.
• 14:30, Lecture Hall H4
  Hugh Thomas (New Brunswick): Exceptional sequences and factorizations of Coxeter elements
  Abstract: The theory of exceptional sequences was developed in the setting of vector bundles on algebraic varieties by Rudakov and his collaborators. The idea was then carried over into the representation theory of hereditary algebras by Crawley-Boevey and Ringel in the early 90's. In the late 90's, Biane, Brady-Watt, and Bessis began to study factorizations of Coxeter elements in reflection groups. I will survey these two notions, and explain their equivalence, which Colin Ingalls and I established in finite and affine type, and which was extended to general hereditary algebras by Igusa-Schiffler. Time permitting, I will also discuss some more recent joint work with Aslak Bakke Buan and Idun Reiten on some special classes of exceptional sequences.
• 16:00, Lecture Hall H4
  Christopher Drupieski (Athens, Georgia): Comparing low degree cohomology for algebraic groups and finite group of Lie type
  Abstract: Let G be a semisimple algebraic group over a field of positive characteristic p, and let G(q) be a finite subgroup of Lie type. In this talk I will discuss conditions under which the cohomology of G with irreducible coefficients is isomorphic to that of G(q). As an application, we can compute the first and second cohomology groups of G(q) under very mild restrictions on p and q when the coefficient module has fundamental highest weight. Some partial results and calculations for algebraic groups of Type C will also be presented. This work is in collaboration with the University of Georgia VIGRE Algebra Group.
• 17:15, Lecture Hall H4 (30 minutes)
  Lutz Hille (Münster): Coxeter Groups and Mutations (jt. with Jürgen Müller, Aachen)
  Abstract: This talk is motivated by the work with Beineke and Brüstle on cluster mutations for quivers with three vertices. We try to understand the group generated by cluster mutations, respectively mutations of exceptional sequences, as a group of linear operators in a vector space of dimension $n(n-1)/2$ (where $n$ is the number of vertices in the quiver, respectively the number of elements in the exceptional sequence). This approach gives, using some results in Bourbaki, a precise answer in some cases. To obtain results for mutations of exceptional sequences we have to modify our approach slightly. The basic idea is to consider the action on a polynomial ring and to derive it. In this way we get matrices of order two acting on a vector space of dimension $n(n-1)/2$ and we can determine the structure of the subgroup generated by these matrices.

Friday, 03 June 2011

• 13:15, Lecture Hall H4
  Zhi-Wei Li (Bielefeld): Derived minimal dg algebras
  Abstract: We call a dg algebra derived minimal, if its derived category of dg-modules has no proper localizing subcategories. This is motivated by the work of Benson, Iyengar and Krause on stratification of triangulated categories. In this talk, I will construct some derived minimal dg algebras, by which we can describe the minimal localizing subcategories of the homotopy category K(InjkG) of injective kG-modules for the Klein four group.
• 14:30, Lecture Hall H4
  Shengkui Ye (Singapore): An application of representation theory to the computation of algebraic K-groups of group rings
  Abstract: In this talk, an injection of homology of groups into algebraic K-groups of group rings is presented. An emphasis will be made on the part of representation theory. If time allowed, I will talk about Artin's theorem, Swan's theorem, Mackey functors and their relations with modules over a category, assembly map in algebraic K-theory and so on.
• 16:00, Lecture Hall H4
  Chris Brav (Hannover): Ping-pong and exceptional vector bundles
  Abstract: We present a strategy for proving that full exceptional collections of vector bundles on projective n-space can be constructed from a standard collection of line bundles, reducing the question of constructibility to the problem of freeness of a certain finitely generated linear group. We use the ping-pong lemma of Fricke-Klein to solve this problem in low dimensions, thus providing a new proof of constructibility of exceptional collections in some cases. We expect a similar ping-pong argument to give constructibility on projective n-space and on some other Fano varieties of Picard rank one. This is joint work in progress with Hugh Thomas.

Friday, 27 May 2011

• 13:15, Lecture Hall H10
  Srikanth Iyengar (Lincoln): The evaluation map and stable cohomology
  Abstract: In my talk I will describe a new approach to the study of certain "evaluation" maps in local algebra and rational homotopy theory, through Tate-Vogel cohomology, and present some applications to computing Bass series. This is work in progress in collaboration with Luchezar Avramov.
• 14:30, Lecture Hall H10
  Dave Benson (Aberdeen): Vector bundles on projective space
  Abstract: I shall talk around the definition of Chern classes for coherent sheaves on projective space, and give a short proof of the Hirzebruch-Riemann-Roch formula in this context.
• 16:00, Lecture Hall H10
  Dan Zacharia (Syracuse): A proof of the strong no loop conjecture
  Abstract: The strong no loop conjecture stated that if S is a simple module over a finite dimensional algebra such that Ext^1(S,S) is not zero, then S must have infinite projective dimension. I will present a proof due to Igusa, Liu and Paquette.

Friday, 13 May 2011

• 13:15, Lecture Hall H10
  Fei Xu (Barcelona): Cohomology of transporter category algebras and support varieties
  Abstract: Support variety theory is traced back to Quillen's work on the spectra of equivariant cohomology rings. Let X be a suitable G-space for a finite group G and k an algebraically closed field of positive characteristic, with the property that H*_G(X,k) is Noetherian. Quillen studied the maximal ideal spectrum V_{G,X}, an affine variety, of H*_G(X,k) and proved that it is a disjoint union of locally closed irreducible affine subvarieties. When X is a point fixed by G, the equivariant cohomology ring reduces to the group cohomology ring H*(G)=Ext*_kG(k,k). Since, to each finitely generated kG-module M, it is possible to consider Ext*_kG(M,M), Carlson was able to define the (cohomological) support variety for M as a subvariety of V_G=V_{G,point}, determined by the kernel of a canonical map Ext*_kG(k,k) --> Ext*_kG(M,M) . It marks the starting point of the support variety theory.
  
  However, the general form of Quillen stratification has been forgotten as one does not see it anywhere apart from Quillen's 1971 papers. One of the reasons seems to be that there did not exist an algebraic construction of H*_G(X) and modules on which H*_G(X) could act. In this talk we examine the case where X=BP, the classifying space of a finite G-poset P. Let GxP be the transporter category and k(GxP) the transporter category algebra. We show H*_G(BP)=Ext*_k(GxP)(\k,\k), where \k is the trivial module of the transporter category. if \m is a finitely generated k(GxP)-module, there exists a natural action of Ext*_k(GxP)(\k,\k) on Ext*_k(GxP)(\m,\m). We can prove that Ext*_k(GxP)(\m,\n) is finitely generated over Ext*_k(GxP)(\k,\k), for any two finitely generated modules, and hence we can develop a support variety theory. Particularly when P is the trivial G-poset, our theory reduces to the one introduced by Carlson. It is interesting to mention that every block algebra of a transporter category algebra is Gorenstein, and the support variety theory of Snashall-Solberg, based on Hochschild cohomology rings, works perfectly for such a block algebra. Furthermore the Snashall-Solberg theory on the "principal block" of a transporter category algebra is closely related to our support variety theory.
  
  If time permits, we shall discuss some properties of the support varieties, including Quillen stratification for modules.
• 14:30, Lecture Hall H10
  Paul Smith (Seattle): Penrose tilings of the plane and noncommutative algebraic geometry
  Abstract: The space X of Penrose tilings of the plane has a natural topology on it. Two tilings are equivalent if one can be obtained from the other by an isometry. The quotient topological space X/~ is bad: every point in it is dense. The doctrine of non-commutative geometry is to refrain from passing to the quotient and construct a non-commutative algebra that encodes some of the data lost in passing to X/~. In this example (see Connes book for details) the relevant non-commutative algebra is a direct limit of products of matrix algebras. We will obtain this non commutative algebra by starting with a certain quotient of the free algebra on two variables treated as the homogeneous coordinate ring of a non-commutative curve. This is similar to treating the preprojective algebra of a wild hereditary algebra as the homogeneous coordinate ring of a non-commutative curve. The category of quasi coherent sheaves on this non-commutative curve is equivalent to the module category over a simple von Neumann regular ring. That von Neumann regular ring is the same as the direct limit algebra that Connes associates to X/~. We will discuss algebraic analogues of various topological features of X/~. For example, the non-vanishing of extension groups between simple modules is analogous to the fact that every point in X/~ is dense (which is equivalent that any finite region of one Penrose tiling appears infinitely often in every other tiling).
  

Sunday, 08 May 2011

• Workshop on Matrix Factorizations

Saturday, 07 May 2011

• Workshop on Matrix Factorizations

Friday, 06 May 2011

• Workshop on Matrix Factorizations

Friday, 15 April 2011

• Matthias Warkentin (Chemnitz): Talk cancelled
• 13:15, Lecture Hall H10
  Greg Stevenson (Bielefeld): Tensor actions and local complete intersections
  Abstract: We will discuss recent progress on understanding the structure of singularity categories associated to noetherian rings. These results are obtained using the formalism of actions by tensor triangulated categories. After giving a brief overview of this formalism we will demonstrate how actions by derived categories allow one to deduce a classification of thick subcategories for the singularity category of a local complete intersection.
• 15:15, Lecture Hall H10
  Gil Kaplan (Tel Aviv): Nilpotency, solvability and the twisting function of finite groups
  Abstract: For a finite group we define the twisting function and study the relations between properties of the twisting function and properties of the group. In particular, we obtain new characterization of the Fitting subgroup, new characterization of nilpotent groups and a sufficient condition for solvability.

Friday, 25 March 2011

• 13:15, Lecture Hall H8
  Helmut Lenzing (Paderborn): Examples illustrating Orlov's theorem

Friday, 11 March 2011

• 13:15, Lecture Hall H8
  Philipp Lampe (Bielefeld): Orlov's Theorem
  Abstract: In this talk, Orlov's description of the stable derived category of a graded Gorenstein algebra will be explained.
• 14:30, Lecture Hall H8
  Jesse Burke (Bielefeld): The singularity category of an affine complete intersection via non-affine matrix factorizations
  Abstract: We introduce a category of non-affine "matrix factorizations" for the section of a line bundle over a noetherian, separated scheme. This category has the same objects as a category recently introduced by Polishchuk-Vaintrob. We show how these non-affine matrix factorizations and a theorem of Orlov can be used to describe the singularity category of a complete intersection ring. This construction is motivated by a desire to extend results of Dyckerhoff and Polishchuk-Vaintrob from hypersurfaces to complete intersections of arbitrary codimension. If time permits we also show how this construction allows generalizations of known results of Avramov-Buchweitz and Avramov-Iyengar from local complete intersections to all complete intersections. This is joint work with Mark Walker.
• 15:45, Lecture Hall H8
  Alastair King (Bath): Grassmannian cluster algebras
  Abstract: I will explain work in progress on a categorification of the homogeneous coordinate rings of Grassmannians, considered as cluster algebras.

Friday, 18 February 2011

• 13:15, Lecture Hall H8
  Martin Bender (Wuppertal): Tilting bundles on crepant resolutions of toric 3-CY-varieties
  Abstract: A brane tiling is a bipartite graph on a torus such that its faces are polygons. Given such a graph one can associate to it a quiver with relations and an affine toric 3-CY-variety. Moduli spaces of certain representations of this quiver then can be seen as crepant resolutions of the affine variety. These moduli spaces depend on the choice of a stability parameter. By varying this parameter we get all toric crepant resolutions. We show how to explicitly determine a tilting bundle (i.e. a vector bundle establishing a derived equivalence between the variety and the endomorphism algebra of this vector bundle) on these spaces. Therefore, perfect matchings on the brane tiling will play the center role. The algorithm arising from this has been suggested in the physical literature.
• 14:30, Lecture Hall H8
  Julian Külshammer (Kiel): Biserial algebras via D_4-subalgebras
  Abstract: Biserial algebras are a class of tame algebras, that frequently occur if one encounters a tame algebra in the representation theory of groups or Lie algebras. In the mid-90s they were described in terms of quivers and relations by W. Crawley-Boevey and R. Vila-Freyer. The path algebra of D_4 appears to be the minimal non-biserial algebra. In this talk we will give a precise criterion for biseriality by looking at subalgebras eAe, where e is an idempotent of A.
  
• 15:45, Lecture Hall H8
  Daiva Pučinskaitė (Kiel): Bases of 1-quasi-hereditary algebras
  Abstract: The topic of this talk is a subclass of quasi-hereditary basic algebras, called 1-quasi-herediditary (roughly speaking, any "possible" multiplicity in a Jordan-Hölder-filtration of the standard modules resp. in a $\Delta$-good filtration of projective indecomposable modules is exactly one). Each of these algebras has a particular basis, which can be described combinatorially. Moreover, they are related to a class of local self-injective algebras. If the Ringel-dual of a 1-quasi-hereditary algebra is also 1-quasi-hereditary, then we construct a basis, such that every basis element generates a $\Delta$-good module.
  

Friday, 11 February 2011

• 13:15, Lecture Hall H8
  Fernando Muro (Sevilla): On the representability theorems of Brown and Adams
  Abstract: These classical results on algebraic topology have had a great impact in algebraic geometry and representation theory since Neeman discovered that they hold in any triangulated category satisfying some of the formal properties of the stable homotopy category.
  While Brown representability holds for a huge class of triangulated categories, the stronger Adams representability seemed to be confined to triangulated categories with a countable generating subcategory of compact objects.
  In 2005 Rosicky stated a theorem asserting that any well generated triangulated category with a model satisfies a transfinite version of Adams representability. In a 2009 paper, Neeman began to explore striking applications of this result. About the same time, a serious gap in the proof of Rosicky's theorem was discovered.
  In this talk I will survey about these topics and discuss the state of the problem, including recent advances by Bazzoni-Stovicek, Braun-Göbel, Raventos and myself, as well as questions which are still open.

Friday, 28 January 2011

• 13:15, Lecture Hall H8
  Yong Jiang (Bonn): Composition algebras of weighted projective lines
  Abstract: In this talk I will present a recent work of I. Burban and O. Schiffmann (arxiv:1003.4412). This work is a further study on composition subalgebras of Hall algebras of weighted projective lines. It was proved that the composition algebra is a topological subbialgebra of the Hall algebra. And the structure of the torsion part of the composition algebra was studied in details. Some interesting applications to weighted projective lines of domestic and tubular type were given.
• 14:30, Lecture Hall H8
  Hanno Becker (Bonn): Khovanov-Rozansky link homology via maximal Cohen-Macaulay approximations and the combinatorics of Soergel bimodules
  Abstract: In this talk, I will outline a new construction of Khovanov-Rozansky link homology based on commutative algebra and the combinatorics of Soergel bimodules. More precisely, we will replace the homotopy category of matrix factorizations, on which the original construction of Khovanov and Rozansky is based, by the equivalent stable category of maximal Cohen-Macaulay modules over the associated hypersurface and see what the KR-construction looks like on this side. It will turn out that the modules we get can be expressed as maximal Cohen-Macaulay approximations to certain Soergel bimodules. The well-understood combinatorics of these modules then immediately translates into properties of KR link homology.
• 15:45, Lecture Hall H8
  Mustafa Kemal Berktas (Uşak, Turkey): Pure Injectivity in Accessible Categories
  Abstract: In a series of articles, Pedro A. Guil Asensio and Ivo Herzog have extended many well known results on pure injectivity to the category of flat modules over a unitary ring. The category of flat R-modules is an accessible category in the sense of Adamek and Rosicky. In this talk, we will discuss how to extend the results of Guil and Herzog to the accessible categories via functorial morphisms.

Friday, 14 January 2011

• 14:00, Lecture Hall H8
  Philipp Lampe (Bielefeld): Quantum cluster algebras and the dual canonical basis
  Abstract: We consider the positive part of the quantized universal enveloping algebra of the Lie algebra sl_{n+1}. The Poincaré-Birkhoff-Witt theorem provides bases of the algebra as well as bases of subalgebras associated with a Weyl group element w. However, the PBW bases are not canonical since they depend on the choice of a reduced expression for w. Therefore, Lusztig defined the canonical basis. It turns out that the canonical basis is related to Fomin-Zelevinsky's cluster algebras. I focus on the case where w is an element of length 2n associated with a terminal module and explain the connection with Geiß-Leclerc-Schröer's non-quantized cluster algebra.
• 15:15, Lecture Hall H8
  Tobias Dyckerhoff (Yale): Matrix factorization categories of isolated hypersurface singularities
  Abstract: Matrix factorizations have been introduced by Eisenbud in his study of homological algebra over hypersurface singularities. Toen's derived Morita theory can be used to study the noncommutative geometry underlying matrix factorization categories. We will calculate Hochschild invariants and establish properties like smoothness and properness. Finally, based on joint work with Daniel Murfet, we will see how local duality, decorated with the perturbation lemma and Grothendieck residues, provides a Calabi-Yau structure on matrix factorization categories.

Sunday, 12 December 2010

• Workshop: Women in Representation Theory: Selfinjective Algebras and Beyond

Saturday, 11 December 2010

• Workshop: Women in Representation Theory: Selfinjective Algebras and Beyond

Friday, 10 December 2010

• Workshop: Women in Representation Theory: Selfinjective Algebras and Beyond

Friday, 03 December 2010

• 13:15, Lecture Hall H8
  Chrysostomos Psaroudakis (Ioannina): Recollements of Abelian Categories and Homological Dimensions
  Abstract: We consider recollements of abelian categories and we discuss how certain homological properties of the categories involved in a recollement situation are related. In particular we are interested in the behavior of global and representation dimension in this context.
• 14:30, Lecture Hall H8
  Jan Stovicek (Bielefeld): A negative solution to Rosicky's problem
  Abstract: In his monograph from 2001, Neeman suggested to study triangulated categories with infinite coproducts using abelian "approximations". These are certain naturally constructed homological and coproduct preserving functors to abelian categories. One knows that these functors are rarely faithful, but Rosicky proposed in 2005 that one may perhaps always find a full functor of this type. In joint work with Silvana Bazzoni, we prove that this is not possible for the unbounded derived categories of several well understood rings.
• 15:45, Lecture Hall H8
  Adam-Christiaan van Roosmalen (Bonn): Infinite version of cyclic quivers and tubes
  Abstract: In the category of coherent sheaves on a smooth projective line, all the torsion objects are contained in uniserial categories, called tubes. By replacing the projective line by weighted version, a similar statement remains true, but one replaces the tubes by "larger" tubes. In this talk, we will introduce an infinite version of such a tube which will be a new uniserial hereditary category with Serre duality, and show how it occurs in the representation theory of some small categories.

Friday, 26 November 2010

• 13:15, Lecture Hall H8
  Felix Dietlein (Köln): Spectral Properties in Auslander-Reiten-Theory encoded by Preprojective Algebras
  Abstract: The spectral theory of Coxeter matrices has turned out to provide a powerful tool in the representation theory of finite dimensional algebras. The aim of this talk is to illustrate how preprojective algebras aid to understand spectral properties in the hereditary case. We examine the influence of quiver geometry on factorisations of Coxeter polynomials and entries of Coxeter eigenvectors. That way we derive classifications of spectral radii and some explicit constructions in terms of representation theoretical data from recent results on preprojective algebras. We give a short overview of other current approaches to this field and indicate how preprojective algebras may also be helpful in these areas.
• 14:30, Lecture Hall H8
  Michael Barot (UNAM Mexico): News about constructions of Lie algebras
  Abstract: We associate to each positive semi-definite unit form three Lie algebras: one using an approach with generators and relations an analogy to Serre's Theorem. The second as Borcherd algebras using bases for the root spaces and the third is obtained as quotient by a uniquely defined ideal from either of the first two. The latter is an extended affine Lie algebra.

Friday, 19 November 2010

• 13:15, Lecture Hall H8
  Markus Severitt (Bielefeld): Simple Representations of some Generalized Group Schemes of Cartan Type
  Abstract: We work over a field of prime characteristic and consider a class of group schemes which are automorphism group schemes of a truncated polynomial ring. The aim of the talk is to give a parametrization of all simple representations of these group schemes. A subclass of them is of Cartan type Witt. That is, their Lie algebra is a restricted Witt algebra. We use the parametrization of all simple restricted modules for the restricted Witt algebras due to Shen, Holmes, and Nakano in order to obtain one for these group schemes. For the other group schemes, unfortunately, we cannot use the simple restricted modules of the Lie algebra. But nevertheless we are able to generalize the picture of the Witt type case.
• 15:00, Lecture Hall H8
  Martin Kalck (Bonn): Gentle algebras and generalized singularity categories
  Abstract: We study Verdier quotients of bounded derived categories of certain gentle algebras of global dimension 2 modulo the subcategory of $\tau$-invariant complexes. Tilting theory yields a relation to singular projective curves, which allows us to describe these categories completely and explicitly. They turn out to be Hom-finite and representation-discrete.

Friday, 12 November 2010

• 13:15, Lecture Hall H8
  Edward Green (Blacksburg): Projective resolutions
  Abstract: In this talk I will present an algorithmic method for constructing projective resolutions of finite dimensional modules over a finite dimensional quotient of a path algebra. This construction uses the results of my previous two lectures.
• 15:00, Lecture Hall H8
  Andrei Marcus (Cluj-Napoca): Crossed products, Brauer groups and Clifford classes
  Abstract: We introduce an equivalence between central simple strongly G-graded algebras. Such classes are associated in a natural way to absolutely irreducible characters of semisimple G-graded algebras. We study invariants of this equivalence relation, and also the structure of certain representatives of the equivalence classes. We show that G-graded Rickard equivalences defined over small fields preserve Clifford classes associated to characters. These equivalences are compatible with operation on Clifford classes defined in terms of central simple crossed products.

Saturday, 06 November 2010

• Workshop: Computer Algebra in Representation Theory of Algebras

Friday, 05 November 2010

• Workshop: Computer Algebra in Representation Theory of Algebras

Friday, 29 October 2010

• 13:00, Lecture Hall H8
  Edward Green (Blacksburg): Right Gröbner bases, Cohn's theorem and resolutions
• 14:10, Lecture Hall H8
  Andrea Solotar (Buenos-Aires): Homological dimension of quantum generalized Weyl algebras
  Abstract: In this talk I will explain how to compute Hochschild homology and cohomology of the class of quantum generalized Weyl algebras defined in [V. Bavula, Generalized Weyl algebras and their representations. St. Petersbourg Math. J. 4 (1), (1990) pp.71--90], adapting methods from [M. Farinati, A. Solotar, M. Suarez-Alvarez, Hochschild homology and cohomology of generalized Weyl algebras. Ann. Inst. Fourier (Grenoble) 53(2), (2003) pp.465--488].
  Examples of such algebras are the quantum n-th Weyl algebras, the quantum enveloping algebra of sl(2), and subalgebras of invariants of these algebras under finite cyclic groups of automorphisms. I will also relate the results with the global dimension of these algebras.
  This is a joint work with Mariano Suarez-Alvarez and Quimey Vivas.
• 15:40, Lecture Hall H8
  Yong Jiang (Bonn): Hall algebra approach to Drinfeld's presentation of quantum loop algebras
  Abstract: This is a joint work with R. Dou and J. Xiao. The purpose of this work is to understand Drinfeld's presentation of quantum loop algebra via Hall algebra of the category of coherent sheaves on weighted projective lines. Based on O. Schiffmann and A. Hubery's work, we find out certain elements in the double Hall algebra satisfying Drinfeld's relations. Thus as a corollary, we deduce that the double composition algebra is isomorphic to the whole quantum loop algebra in case of finite or affine type.

Friday, 22 October 2010

• 13:15, Lecture Hall H8
  Edward Green (Blacksburg): Basic Tools For Computational Homological Algebra
  Abstract: In this introductory talk, I will discuss the basic tools needed to computationally study finite dimensional algebras and modules. I will assume knowledge of quivers, path algebras and their quotients, representations of quivers and begin with the theory of Groebner bases for path algebras. If time permits, I will extend the theory of Groebner bases for ideals to right Groebner bases for right ideals and vertex projective modules.

Saturday, 09 October 2010

• Workshop: Projective Dimension 2

Friday, 08 October 2010

• Workshop: Projective Dimension 2

Friday, 24 September 2010

• 10:00, Room V2-216
  Hideto Asashiba (Shizuoka): The "derived category" of a lax functor
  Abstract: We fix a commutative ring k and a small category I, and denote by k-Cat the 2-category of k-linear categories. As a generalization of a group action on a k-category, we consider a functor X: I --> k-Cat (when I is a group, this is just a group action). The purpose of the talk is to define its "module category" Mod X and "derived category" D(Mod X) to investigate derived equivalences of those X. If I is not a groupoid, then an expected candidate of the definition does not work within the limits of functors, and it needs to define them as (op)lax functors. Therefore we work over oplax functors, and for each oplax functor X we will define Mod X, D(Mod X) as oplax functors I --> k-Cat.
  The Morita type theorem characterizing derived equivalences by Rickard and Keller will be generalized in this setting.

Friday, 03 September 2010

• 13:15, Room V2-216
  Rasool Hafezi (Isfahan): Homotopy categories arising from quiver representations
  Abstract: In this talk, we investigate representations of quivers by modules over arbitrary rings and look at the corresponding homotopy categories. We discuss compact objects and compact generating sets of these homotopy categories and deduce the existence of some adjoint functors by classifying totally acyclic complexes of projective (injective) representations of certain quivers.
• 14:30, Room V2-216
  Xiaojin Zhang (Nanjing): The existence of maximal orthogonal subcategories
  Abstract: We build a connection between Auslander's 1-Gorenstein algebras of global dimension 2 admitting trivial maximal 1-orthogonal subcategories and tilted algebras of finite type, and we draw the quivers of such algebras. Moreover, all (n-1)-Auslander algebras admitting trivial maximal (n-1)-orthogonal subcategories can be determined.

Friday, 23 July 2010

• 13:15, Room T2-220
  Guodong Zhou (Bielefeld): The Batalin-Vilkovisky structure over the Hochschild cohomology ring of a symmetric algebra
  Abstract: In the 60s of last century, Gerstenhaber discovered a graded Lie algebra structure over the Hochschild cohomology of an associative ring. It is usually very difficult to compute this structure, as his construction is rather complicated. Tradler noticed in 2002 that the Hochschild cohomology ring of a symmetric algebra has a Batalin-Vilkovisky structure, that is, there is a $\Delta$ operator over Hochschild cohomology which together with the cup-product can determine the Gerstenhaber Lie bracket. In a work in progress with Jue Le, we use this idea to compute the Gerstenhaber Lie bracket for some examples, including the group algebras of finite abelian $p$-groups and monogenic algebras. We shall also investigate the behavior of the $\Delta$-operator under the additive decomposition of the Hochschild cohomology of a finite group algebra.
• 14:30, Room T2-220
  Ivo Dell'Ambrogio (Singapore): On support varieties for (G-) C*-algebras
  Abstract: Topological K-theory, and its bivariant version KK-theory, are probably the most useful invariants of C*-algebras and G-C*-algebras (= dynamical systems, for locally compact groups G). For a fixed G, these abelian groups and their operations form a tensor triangulated category, KK^G, which is characterized by a nice universal property.
  
  We explain how geometric ideas imported from other domains of mathematics can be applied to study the global structure of these categories. In particular, for G finite we show how to define a well-behaved theory of support varieties for (sufficiently nice) G-C*-algebras which, for G trivial at least -- and conjecturally for all finite G -- allows the classification of all localizing tensor ideal subcategories.
• 16:00, Room T2-220
  Javad Asadollahi (IPM / Univ. Isfahan): Cohomology theories based on flat modules
  Abstract: The quotient category of the homotopy category of flat modules by its full subcategory consisting of pure complexes, known as the pure derived category of flats, is first studied by A. Neeman. One of the important features of working in this category is that flat resolutions are unique up to homotopy and so can be used to compute cohomology. Our aim in this talk is to review the constructions and properties of Tate and complete cohomology theories in this quotient category.

Friday, 16 July 2010

• 11:00, Room V5-227
  Andrew Hubery (Leeds): Tangent Spaces, Massey Products, and Applications to Representation Theory
  Abstract: We look again at an article by Crawley-Boevey and Schröer where they generalise the notion of canonical decomposition for representations of quivers to arbitrary algebras. The proof is almost entirely elementary, except for one part, where they have to develop some deformation theory. They remark, however, that an elementary approach to this last part is possible using tangent space methods, provided the schemes are (generically) reduced. We show how to get around this problem by extending the standard tangent space techniques.
• 13:15, Room T2-220
  Julia Sauter (Leeds): On quiver flag varieties
  Abstract: We review the quiver-graded Springer correspondence introduced by Markus Reineke. Fibres of the first projection are called quiver flag varieties. Their geometrical properties are still rather unknown. We answer the question when they are orbit finite via looking at a category of modules and using the Happel-Vossieck list.
• 14:30, Room T2-220
  Selene Sanchez (Köln): The graded Lie algebra on the Hochschild cohomology of a modular group algebra
  Abstract: The space of outer derivations of an associative algebra is the quotient of its derivations modulo the inner derivations. It is well known that the commutator provides to this space a Lie algebra structure. We also know that this space is the first group of the Hochschild cohomology of an associative algebra. Furthermore, in 1963, Gerstenharber defined a bracket on the Hochschild cohomology groups of higher degrees which restricted to the first degree is the commutator bracket. Moreover, the Gerstenhaber bracket endows the Hochschild cohomology with a graded Lie algebra structure. In this talk, I will remind the definition of Gerstenhaber bracket. Then I will give some explicit examples of the Lie structure. For instance in the case of the group algebra of the cyclic group over a field of characteristic equal to the order of the group.
• 16:00, Room T2-220
  Grzegorz Bobinski (Torun): The gentle algebras derived equivalent to the cluster tilted algebras
  Abstract: The gentle cluster tilted algebras were described and classified (up to the derived equivalence) in papers by Caldero--Chapoton--Schiffler, Buan--Vatne, Assem--Bruestle--Charbonneau-Jodoin--Plamondon, and Bastian. In a joint work with Buan we classify the gentle algebras which are derived equivalent to the gentle cluster tilted algebras (recall that the gentle algebras are closed under the derived equivalence) An important role in this classification is played by a combinatorial invariant introduced by Avella-Alaminos and Geiss. It is also worth to mention that two algebras from the considered class are derived equivalent if and only if they can be connected via a sequence of Brenner--Butler tilts.

Friday, 09 July 2010

• 13:15, Room T2-220
  Sefi Ladkani (Bonn): Perverse equivalences, mutations and applications
  Abstract: Perverse Morita equivalences are special kind of derived equivalences introduced by Chuang and Rouquier. We shall consider a specific case, describe its relation with the Brenner-Butler tilting modules and explain how it gives rise to the notion of mutations of algebras, which are local operations on algebras producing derived equivalent ones.
  
  We will then relate these operations with another local operation, namely the Fomin-Zelevinsky quiver mutation, and present applications to endomorphism algebras of cluster-tilting objects in 2-Calabi-Yau categories as well as algebras of global dimension two.
• 14:30, Room T2-220
  Dan Zacharia (Syracuse): On Auslander-Reiten components for selfinjective algebras I
  Abstract: see Kerner
• 16:00, Room T2-220
  Otto Kerner (Düsseldorf): On Auslander-Reiten components for selfinjective algebras II
  Abstract: In studying the modules of finite complexity over selfinjective algebras, a class of indecomposable modules called $\Omega$-perfect modules, has played an important role. More precisely, we have shown that every stable Auslander-Reiten component where each module has finite complexity and is eventually $\Omega$-perfect, has a very nice shape. The components not containing such modules (even without assuming finite complexity), also have the same very nice shape. We will talk about these and other related results about $\Omega$-perfect modules.

Friday, 25 June 2010

• 14:30, Room T2-220
  Claudia Köhler (Bielefeld / Paderborn): Thick subcategories of self-injective algebras
  Abstract: The stable module category of a self-injective algebra is endowed with a triangulated structure. Thus, we may ask for the thick subcategories of these categories. I concentrate on the case that the algebra is a self-injective standard algebra of finite representation type. There is an elaborate classification of these algebras by Riedtmann assigning to each self-injective representation-finite algebra a Dynkin type. Asashiba shows that this type determines the algebra up to stable equivalence.
  
  On the other hand, there is a combinatorial classification of the exact abelian extension-closed subcategories of the category of representations of a quiver of Dynkin respectively extended Dynkin type given by Thomas and Ingalls.
  
  In my talk, I will connect these concepts and show how we can use the given classification in the hereditary case to gain similar results for self-injective algebras.
• 16:00, Room T2-220
  Adam-Christiaan van Roosmalen (MPI Bonn): Hereditary categories with Serre duality generated by preprojectives
  Abstract: In their classification of hereditary noetherian categories with Serre duality, Reiten and Van den Bergh discovered a new type of category: a hereditary category not generated by projective objects, but by preprojective objects.  It can be shown that --up to derived equivalence-- these categories are equivalent to the categories of finite dimensional representations of strongly locally finite quivers (i.e. all indecomposable projective and injective representations have finite dimension).
  
  After Ringel has constructed examples to shown that not every hereditary category with Serre duality is derived equivalent to a noetherian one, Reiten suggested whether one could classify hereditary categories with Serre duality generalted by preprojective objects.
  
  We introduce thread quivers as a generalization of quivers without relations. With these new objects, we can classify the nonnoetherian hereditary categories with Serre duality generated by preprojectives: such categories are derived equivalent to categories of finitely presented representations of thread quivers.  In particular, thread quivers generalize the nonnoetherian examples given by Ringel.

Friday, 18 June 2010

• 13:15, Room T2-220
  Xiao-Wu Chen (Bielefeld / Paderborn): Compact generators in categories of matrix factorizations, after Dyckerhoff
  Abstract: This is a report on a paper by Tobias Dyckerhoff (arXiv:0904.4713v4).
• 14:30, Room T2-220
  Janine Bastian (Hannover): Derived equivalences for cluster-tilted algebras of Dynkin type D
  Abstract: This is a joint work with Thorsten Holm and Sefi Ladkani.
  
  Cluster-tilted algebras arise as endomorphism algebras of cluster-tilting objects in a cluster category. For the special case of cluster categories of Dynkin quivers the cluster -tilted algebras are known to be of finite representation type.  Moreover, by a result of Buan, Marsh and Reiten they can be described as quivers with relations by a simple combinatorial recipe. As a consequence, a cluster-tilted algebra of Dynkin type is uniquely determined by its quiver.
  
  In this talk we give a complete good mutation equivalence classification of cluster-tilted algebras of Dynkin type D. Moreover, we give a far reaching derived equivalence classification and suggest explicit normal forms for the derived equivalence classes.
  
• 16:00, Room T2-220
  Steffen Oppermann (Köln): Stable categories of higher preprojective algebras
  Abstract: In this talk I will introduce the notion of n-representation finiteness, generalizing hereditary representation finite algebras. Inspired by the hereditary case (n=1) I will introduce higher preprojective algebras, and show that for n-representation finite algebras they are finite dimensional and self-injective.
  
  Hence it makes sense to study the stable module categories of these higher preprojective algebras. I will show that these stable module categories are (n+1)-Calabi-Yau. It will be indicated how they can be identified with the (n+1)-Amiot cluster category of the stable n-Auslander algebras of the n-representation finite algebras.
  
  Finally I will point out that the above results still hold if the assumption of n-representation finiteness is weakened to a less restrictive vanishing property, which in particular always holds for n = 2. In this case the higher preprojective algebra will not be self-injective, but Gorenstein of dimension at most 1. In the formulation of results above one then has to replace the stable module category by the stable category of Cohen-Macaulay modules.

Friday, 04 June 2010

• Reading Course Cluster Algebras
  13:15, Room T2-220
  Reiner Hermann (Bielefeld): Acyclic Calabi-Yau categories
• Reading Course Cluster Algebras
  14:30, Room T2-220
  Xiao-Wu Chen (Bielefeld / Paderborn): Cluster categories of canonical algebras
• 16:00, Room T2-220
  Claire Amiot (Bonn): Algebras of cluster type $\tilde{A}_n$
  Abstract: This is a joint work with Steffen Oppermann.
  We say that an algebra A of global dimension at most two is of cluster type Q, where Q is an acyclic quiver, if the generalized cluster category  C_A is equivalent to the cluster category C_Q. For example any tilted algebra of global dimension 2 and of type Q is of cluster type Q. However the converse is not true.
  In this talk, I will explain how it is possible to classify all derived equivalence classes of algebra of cluster type Q using mutation of graded quiver with potential.
  I will then focus on Q being an orientation of $\tilde{A}_n$, which is the most simple non trivial case, and give a very explicit description of all algebras of cluster type Q.

Friday, 21 May 2010

• Reading Course Cluster Algebras
  13:15, Room T2-220
  Phillip Linke (Bielefeld): Mutation in triangulated categories
• Reading Course Cluster Algebras
  14:30, Room T2-220
  Claudia Köhler (Paderborn / Bielefeld): Triangulated orbit categories
• 16:00, Room T2-220
  David Ploog (Hannover): Exceptional collections, spherical twists and Coxeter functors
  Abstract: In various areas of geometry and algebra, Coxeter elements (a notion of lattice theory) have been lifted to triangulated categories. We present two such approaches, and show how to compare them.

Friday, 30 April 2010

• Reading Course Cluster Algebras
  13:15, Room T2-220
  Zhi-Wei Li (Bielefeld): Cluster-tilting in a general framework
• Reading Course Cluster Algebras
  14:30, Room T2-220
  Zhe Han (Bielefeld): From triangulated categories to cluster algebras
• 16:00, Room T2-220
  Andrei Zelevinsky (Boston): Cluster algebras via quivers with potentials
  Abstract: Based on a joint work with Harm Derksen and Jerzy Weyman, we will discuss a setup for cluster algebras in terms of quivers with potentials and their decorated representations. This setup allows us to prove several conjectures about F-polynomials and g-vectors for cluster algebras with arbitrary skew-symmetric exchange matrices.

Friday, 23 April 2010

• Reading Course Cluster Algebras
  13:15, Room T2-220
  Guodong Zhou (Paderborn): Cluster categories
• Reading Course Cluster Algebras
  14:30, Room T2-220
  Jue Le (Paderborn): Cluster-tilted algebras
• 15:45, Room T2-220
  Matthias Künzer (Aachen/Stuttgart): From n-triangles to Heller triangulated categories (after S. Thomas)
  Abstract: In Heller triangulated categories, 2-triangles are just Verdier triangles, whereas 3-triangles are particular Verdier octahedra. More generally, there are n-triangles which enjoy the same properties as Verdier triangles: one can form cones, rotate, prolongate morphisms (nonuniquely). These n-triangles form a simplicial set. The usual way to specify these n-triangles is: define n-pretriangles by exactness properties, pick an isomorphism theta_n between induced shift functors and let an n-triangle be an n-pretriangle X such that X theta_n = id. Sebastian Thomas discovered that this process can be reversed: picking a subset {n-triangles} in {n-pretriangles} satisfying some properties, just as Verdier did for n = 2, one can recover theta_n. This facilitates the access to Heller triangulated categories. While theta_n and {n-triangles} determine each other, sometimes theta_n is easier to handle; e.g. in showing that an adjoint of an exact functor is exact.
• 17:00, Room T2-220
  Changchang Xi (Beijing): Derived equivalences and cohomological approximations
  Abstract: In this talk, we shall provide an abstract method to construct derived equivalences between \Phi-Auslander-Yoneda algebras from an almost D-split triangle with certain cohomological conditions in a triangulated category, where \Phi is an admissible subset of the natural numbers. The method can be applied to the derived categories of rings, Calabi-Yau categories and Frobenius categories. This is a part of my joint work with Hu and Koenig.

Friday, 16 April 2010

• Reading Course Cluster Algebras
  13:15, Room T2-220
  Estanislao Herscovich (Paderborn): Cluster algebras I
• Reading Course Cluster Algebras
  15:00, Room T2-220
  Karsten Dietrich (Bielefeld): Cluster algebras II

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