Seminar
No talks have been announced for this week.
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Seminar Archive
Friday, 20 July 2018
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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.
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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
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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
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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).
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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
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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
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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.
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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
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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.
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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
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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)
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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
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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.
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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
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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
Friday, 20 April 2018
Friday, 13 April 2018
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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
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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.
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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
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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.
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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
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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.
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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.
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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
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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.
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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.
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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
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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
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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
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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
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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)
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BiBo Seminar
13:45, Room V5-227
Andrew Hubery (Bielefeld): Preprojective algebras revisited
Friday, 03 November 2017
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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
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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
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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.
For information on earlier talks please check the complete seminar archive.
