Seminar
Friday, 17 November 2017

14:15, Room T2213
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 (noncommutative) finite dimensional algebras the meaning is less clear.
The aim of my talk is to investigate when ringmorphisms induce functors between singularity categories (and related cocomplete categories). One may hope that this gives some idea what information survives in the singularity category.
For a regular email announcement please contact birep.
Future Talks
Friday, 24 November 2017

14:15, Room T2213
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, 08 December 2017

Room T2213
Michael Wong (Austin): tba
Friday, 15 December 2017

Room T2213
Oliver Lorscheid (Rio de Janeiro): tba

Room T2213
Julian Külshammer (Stuttgart): tba
Seminar Archive
Thursday, 09 November 2017

BiBo Seminar
12:15, Room V5227
Magdalena Boos (Bochum): The algebraic UQuotient of the nilpotent cone
Abstract: We consider the conjugationaction of the standard unipotent subgroup U of GL_n(C) on the nilpotent cone N of complex nilpotent matrices of squaresize 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 Uorbits in N, define a set of Uinvariants which span C[N]^U and use these concepts to generically separate the Uorbits. 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 V5227
Andrew Hubery (Bielefeld): Preprojective algebras revisited
Friday, 03 November 2017

14:15, Room T2213
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 bottomup with a topdown approach for its determination. We compare the functorial properties of the AuslanderReiten 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 T2213
Jeanne Scott (Bogotá): Towards a JucysMurphy theory for the Okada algebras
Abstract: I'll discuss work in progress which aims to construct JucysMurphy elements in the Okada algebra F_n together with a corresponding notion of content for the YoungFibonacci lattice which encodes the spectrum of the JucysMurphy elements with respect to the FibonacciTableau bases for irreducible F_nrepresentations.
Friday, 13 October 2017

14:15, Room T2213
Alexander Samokhin (Düsseldorf): Tstructures 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 pushforward and pullback 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 nonstandard tstructures on the derived categories of flag varieties. These tstructures 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 V2200
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 V2200
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 V2200
David Ploog (Berlin): Exceptional sequences for the Auslander algebra of the fat point
Abstract: This algebra is wellknown in representationtheory. 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
XiaoWu 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^2stable tilting sheaves and the one of gstable 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): CohenMontgomery 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 kcategories with projective Homspaces. Let R, S be categories with Gactions and A, B Ggraded categories such that there exist Gcovering functors R —> A and S —> B. Few years ago we established an equivalence between Ginvariant SRbimodules and Ggraded BAbimodules, an analogue of the socalled CohenMontgomery duality, and as an application gave oneone correspondence between Ginvariant stable equivalence of Morita type between R and S and Ggraded 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 BrauerThrall 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 Amodules as soon as A is a representationinfinite finitedimensional algebra over an algebraically closed field. This is in a sense the strongest possible version of the second BrauerThrall 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 BrauerThrall 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: AuslanderBuchweitz 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 Extfunctor 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
XiaoWu Chen (Hefei): Introducing Kstandard additive categories
Abstract: We introduce the notion of Kstandard 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 Rmodules (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 Rmodules 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 finitedimensional algebras and quiver Grassmannians. In both cases we construct desingularizations assuming the module is genfinite, 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 SauterCrawleyBoevey.

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 padic 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 Extquiver of A are fulfilled. If the Extquiver 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 1torsion (= 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 AuslanderGrusonJensen 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 nspace
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 socalled “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 RingelHall algebras and Drinfeld double
Abstract: Inspired by the works of Bridgeland and Gorsky, we define an algebra from the RingelHall algebra of the category formed by Z/2graded complexes over a hereditary abelian category which may not have enough projective objects. Such algebra is called modified RingelHall 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/2graded complexes. The second one is that in twisted case it is isomorphic to the Drinfeld double RingelHall algebra of the hereditary abelian category. Finally, if the category has a tilting object T, then its modified RingelHall algebra is isomorphic to the Z/2graded semiderived Hall algebra and also the Bridgeland's RingelHall 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 Extorthogonal 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 Extorthogonal 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 semiorthogonal decompositions
Abstract: Motivated by the local flavor of several wellknown semiorthogonal 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, blowups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semiorthogonal decompositions for BrauerSeveri varieties.
This is joint work in progress with Daniel Bergh.
Friday, 28 April 2017

14:15, Lecture Hall H10
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.

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 nonalgebraically 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 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.

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 cohX be the category of coherent sheaves over a weighted projective line X and let D^b(cohX) 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(cohX) 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 treeshaped basis of its group of selfextensions, one can associate an affine space of representations with the same dimension vector. If all representations in this cell are indecomposable and pairwise nonisomorphic, 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 nonisomorphic 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 U2113
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.
For information on earlier talks please check the complete seminar archive.