Seminar
Tuesday, 28 February 2017

14:15, Room V2213
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 socalled 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.
For a regular email announcement please contact birep.
Future Talks
Thursday, 09 March 2017

16:15, Room U2135
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 3fold 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.
Friday, 07 April 2017

Robert Marsh (Leeds): Rigid and Schurian modules over tame clustertilted algebras
Abstract: We classify the indecomposable rigid and Schurian modules over a clustertilted algebra of tame representation type. Such a clustertilted 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 Bmodule. 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 Bmodules.
The talk is based on joint work with Idun Reiten.
Friday, 21 April 2017

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 finitedimensional 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 ArtinSchelter regular (Z)algebras, and how one can use their classification to compute the Hochschild cohomology of all finitedimensional 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.

Thorsten Weist (Wuppertal): tba
Friday, 28 April 2017

ChunJu 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 fixedpoint subalgebra associated to an involution.
This is a joint work (arXiv:1609.06199) with Z. Fan, Y. Li, L. Luo, and W. Wang.
Friday, 05 May 2017

Jorge Vitória (London): tba
Seminar Archive
Friday, 03 February 2017

14:15, Room U2135
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 finitedimensional 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 clustertilted algebras studied by GeißLeclerc–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 U2135
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 AuslanderReiten quiver, we study the distance W(C) between the two nonintersecting 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 quasirank 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 U2135
Kevin De Laet (Antwerpen): The connection between Sklyanin algebras and the finite Heisenberg groups
Abstract: The 3dimensional 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 C01226
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 Kalgebras, 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 U2135
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 selfinjective 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 quasihereditary structure on the Auslander algebra if and only if our original algebra is uniserial.

14:30, Room U2135
Florian Gellert (Bielefeld): Maximum antichains in subrepresentation posets
Abstract: For indecomposable representations of Dynkin quivers, the structure of indecomposable morphisms is given by the AuslanderReiten 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 U2135
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 nonselfinjective 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 2n2, 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 representationfinite 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 V2210/216
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 V2210/216
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 finitedimensional hereditary algebra over a finite field. Moreover, we prove that this number is positive whenever the Grassmannian is nonempty, 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 nonnegative coefficients.

NWDR Workshop Winter 2016
14:00, Room V2210/216
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 ChurchFarb. 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 V2210/216
PierreGuy 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 V2210/216
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 multiplicityfree 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 U2135
Matthew Pressland (Bonn): Dominant dimension and canonical tilts
Abstract: Any finite dimensional algebra with dominant dimension d admits a 'canonical' ktilting 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 representationfinite algebra A, CrawleyBoevey and Sauter (generalising Cerulli Irelli, Feigin and Reineke) used the tilt B_1 to construct desingularisations of certain varieties of Amodules. More generally, for d at least 2, any algebra of dominant dimension d is the endomorphism algebra of a generatingcogenerating 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 dAuslander (or, more generally, dAuslander–Gorenstein) algebras. This is joint work with Julia Sauter.
Friday, 18 November 2016

14:15, Room U2135
Sondre Kvamme (Bonn): A generalization of finitedimensional IwanagaGorenstein algebras
Abstract: We will introduce certain wellbehaved 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 finitedimensional algebra. We will also define what it means for the comonad to be Gorenstein, and state analogues of some classical results for IwanagaGorenstein algebras. We will illustrate the constructions and results on specific examples.

15:30, Room U2135
Rosanna Laking (Bonn): KrullGabriel dimension, Ziegler spectra of module categories and applications to compactly generated triangulated categories.
Abstract: We will begin by defining the notion of KrullGabriel (KG) dimension for the module category of a ring R (with many objects) and outlining how this relates to a topology on the indecomposable pureinjective Rmodules. Using examples, we aim to explain how the KGdimension "measures" transfinite factorisations of morphisms in Rmod.
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 deriveddiscrete algebra A and calculate its KGdimension. 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 U2135
Shengfei Geng (Chengdu): Tilting modules and support tautilting modules over preprojective algebras associated with symmetrizable generalized Cartan matrices
Abstract: For each skewsymmetrizable generalized Cartan matrix, GeissLeclecSchröer 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 nonDynkin type. Based on this, we proved that there is a bijection between the sets of support tautilting 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 U2135
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 1Gorenstein 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 clustertilted algebras and representations of quivers over local rings.
Friday, 28 October 2016

13:15, Room U2135
Gustavo Jasso (Bonn): Mesh categories of type Ainfinity and tubes in higher AuslanderReiten theory
Abstract: This is a report on joint work with Julian Külshammer. We construct higher analogues of mesh categories of type Ainfinity and of the tubes from the viewpoint of Iyama's higher AuslanderReiten 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 U2135
Julian Külshammer (Stuttgart): Spherical objects in higher AuslanderReiten 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 mCalabiYau triangulated category with almost split triangles. Moreover, its AuslanderReiten quiver has m1 connected components of type ZAinfinity. Building upon work of Amiot, Guo, Keller, and OppermannThomas, for each positive integer d we construct an mdCalabiYau (d+2)angulated category with almost split (d+2)angles. Moreover, its higher AuslanderReiten quiver has m1 connected components of higher mesh type Ainfty. For m=2, our construction is analogous to the cluster a category of type Ainfinity introduced by HolmJørgensen.

16:00, Room U2135
SvenAke 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 socalled quasiabelian categories. Indeed, every quasiabelian category, in particular the category of Banach spaces, is derived equivalent to the (abelian) heart of the canonical tstructure 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 U2135
Michael K. Brown (Bonn): Topological Ktheory of dg categories of graded matrix factorizations
Abstract: Topological Ktheory of complexlinear dg categories is a notion recently introduced by A. Blanc. The main goal of the talk is to discuss a calculation of the topological Ktheory of the dg category of graded matrix factorizations associated to a complex quasihomogeneous 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 V5227
ZhiWei Li (Xuzhou): A homotopy theory of additive category with suspensions
Abstract: We give a definition of partial onesided triangulated categories. We show that complete cotorsion pairs in exact categories, torsion pairs and mutation pairs in triangulated categories all extend to partial onesided triangulated categories. We prove that partial onesided triangulated categories yield onesided 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 KoenigZhu, KellerReiten and KussinLenzingMeltzer. The second one is the construction of stable triangulated categories which allows us to model IyamaYoshino subfactors of triangulated categories via Quillen closed model structures. The last one is to develop a homotopy theory of additive categories with suspensions via GabrielZisman localization which leads to a Buchweitz type theorem in triangulated categories. This theorem extends the recent work of Wei and IyamaYang which are generalizations of Buchweitz's work on singularity categories. As a corollary we give a triangle equivalence between Verdier quotients and IyamaYoshino subfactors of triangulated categories under suitable conditions.
Thursday, 21 July 2016

16:15, Room V3204
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^2pqr, 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 C01142
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 WedderburnArtin 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 socalled 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 C01142
Martin Kalck (Edinburgh): Knörrertype equivalences for twodimensional cyclic quotient singularities
Abstract: We construct triangle equivalences between singularity categories of twodimensional 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 quasihereditary algebras. This is based on joint work with Joe Karmazyn.

16:00, Room C01142
Grzegorz Bobiński (Torun): On nonsingularity in codimension one of irreducible components of module varieties over quasitilted 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 quasitilted algebras.
Friday, 01 July 2016

13:15, Room C01142
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 RingelSchmidmeier, 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 C01142
Alexander Kleshchev (Eugene): Stratifications of KhovanovLaudaRouquier algebras
Abstract: We review standard module theory for KhovanovLaudaRouquier 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 C01142
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 C01142
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 C01142
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 C01142
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 C01142
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 C01142
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 C01142
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 LittlewoodRichardson 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 C01142
Andreas Hochenegger (Köln): Spherical subcategories
Abstract: In a triangulated category, a spherical object is defined as a CalabiYau object that has a twodimensional (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 CalabiYau property, illustrated by examples.
This is joint work with Martin Kalck and David Ploog.

14:30, Room C01142
David Ploog (Berlin): Discrete triangulated categories
Abstract: The study of discretederived 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 tstructures 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 C01142
RagnarOlaf Buchweitz (Toronto): Tilting theory for onedimensional Gorenstein algebras
Abstract: We show that for a connected, commutative, positively graded Gorenstein algebra R of Krull dimension one wth nonnegative ainvariant 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 CohenMacaulay 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 C01142
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 PerfX embeds as an admissible subcategory into PerfM_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 C01142
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 C01142
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 nonsimple 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 AuslanderReiten 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 C01142
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 C01142
Oriol RaventosMorera (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 C01142
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 C01142
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 LuntsOrlov'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 quasicoherent 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 V3201
Chrysostomos Psaroudakis (Trondheim): Realisation Functors in Tilting Theory
Abstract: Let T be a triangulated category and H the heart of a tstructure 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 tstructure, BeilinsonBernsteinDeligne 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).
For information on earlier talks please check the complete seminar archive.