Seminar
No talks have been announced for this week.
For a regular email announcement please contact birep.
Future Talks
Friday, 13 October 2017

Alexander Samokhin (Düsseldorf): tba
Friday, 20 October 2017

Jeanne Scott (Bogotá): tba
Thursday, 09 November 2017

BielefeldBochum Seminar
Andrew Hubery (Bielefeld): tba
Seminar Archive
Grzegorz Bobinski made his notes from some of the seminar talks available on his web page.
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.
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.
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.
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).
Wednesday, 09 March 2016

10:15, Room V2213
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 LabardiniFragoso. 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
Tuesday, 01 March 2016
Friday, 05 February 2016

14:15, Room V2213
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 V2210/216
Paul Balmer (Los Angeles): An invitation to tensortriangular geometry
Abstract: We will begin by an overview of the various fields where tensortriangulated 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 V2213
Rebecca Reischuk (Bielefeld): The adjoints of the Schur functor
Abstract: The socalled 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 wellknown 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 V2213
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 V2213
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 V2213
Matthew Pressland (Bath): Internally CalabiYau algebras and clustertilting objects
Abstract: Cluster categories, which are 2Calabi–Yau triangulated categories containing clustertilting 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 2CalabiYau, although such a categorification is only known in a few cases. It is observed by KellerReiten that the endomorphism algebra of a clustertilting object in such a category has a certain relative, or internal, CalabiYau symmetry. In this talk, I will explain how to go in the opposite direction; given an algebra A with a suitable level of CalabiYau symmetry, I will explain how to construct a Frobenius category admitting a clustertilting object with endomorphism algebra A.

15:30, Room V2213
Igor Burban (Cologne): Singular curves and quasihereditary algebras
Abstract: In my talk (based on a joint work with Yu. Drozd and V. Gavran), I shall describe a certain noncommutative 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(Amod) as a full subcategory.
Friday, 15 January 2016

14:15, Room V2213
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 quasicoherent 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 V2213
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 nonprime cluster variables.

15:30, Room V2213
Henning Krause (Bielefeld): The variety of subadditive functions for finite group schemes
Abstract: For a finite group scheme G, Friedlander and Pevtsova introduced pipoints which give rise to certain endofinite 'point modules'. Using then CrawleyBoevey'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 V2213
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 V2213
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 twosided 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
Friday, 04 December 2015
Friday, 27 November 2015

14:15, Room V2213
Hans Franzen (Bonn): DonaldsonThomas 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 V2213
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 skewsymmetric cluster algebras with coefficients. As a consequence we obtain a description of the category of CohenMacaulay modules over certain isolated singularities as the completed orbit category of a Nakajima category.

14:30, Room V2213
Magnus Engenhorst (Bonn): Maximal green sequences for quiver categories
Abstract: Maximal green sequences were introduced as combinatorical counterpart for DonaldsonThomas invariants for 2acyclic 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 tstructures 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 V2213
Raymundo Bautista (Morelia): Differential tensor algebras, boxes, and exact structures
Abstract: A box B consists of an algebra A over some field, an AA bimodule U with a coassociative comultiplication, and a counit. 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 tamewild dichotomy proved by Y. Drozd and the discovery due to W. CrawleyBoevey 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 AA bimodule given by the kernel of the counit (an AA 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 quasihereditary algebras and the category of modules with standard filtration.
Friday, 13 November 2015

14:15, Room V2213
Thomas Gobet (Kaiserslautern): On twisted filtrations on Soergel bimodules
Abstract: The IwahoriHecke 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 KazhdanLusztig positivity.
Friday, 06 November 2015

14:15, Room V2213
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 conjugationaction 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, Deltafiltrations and more. (This is work in progress, joint with M. Bulois)

15:30, Room V2213
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 nontrivial psubgroups. Some time will be spent introducing the Brown complex and related basic questions.
Friday, 30 October 2015

14:15, Room V2213
Dirk Kussin (Paderborn): Infinitedimensional 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 pureinjective modules.

15:30, Room V2213
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 Homfinite 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 adbc=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 quasicoherent sheaves (resp. indecomposable infinite dimensional Bmodules) of irrational slope r, a problem attacked by Harland and Prest during the last years through modeltheoretic methods.
Friday, 23 October 2015

14:15, Room V2213
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 RingelSchmidmeier 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 V2213
Julia Sauter (Bielefeld): On quiver Grassmannians and orbit closures for representationfinite algebras
Abstract: We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projectiveinjective; its endomorphism ring is called the projective quotient algebra.
For any representationfinite 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 CrawleyBoevey 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 pexact categories
Abstract: This is a survey talk on homological algebra in a nonadditive 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, LefevreHasegawa and Positselski, to the case of Ainfinity 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 spherelike objects.
Friday, 12 June 2015

14:00, Room U2113
Philipp Lampe (Bielefeld): Almost periodic sequences attached to noncrystallographic root systems
Abstract: We study FominZelevinsky’s mutation rule in the context of noncrystallographic 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 U2113
Lutz Hille (Münster): Derived equivalences between GLweighted 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 (instantonless 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 V3201
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 GromovWitten 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 E0180
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 E0180
Sarah Scherotzke (Bonn): Quiver varieties and selfinjective 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 finitelygenerated algebra P, which is selfinjective 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 finitelygenerated 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 selfinjective algebras of finite type.
Friday, 15 May 2015

13:15, Room T2233
Raf Bocklandt (Amsterdam): Gentle A_infinity Algebras and Mirror Symmetry
Abstract: We introduce the notion of gentle Ainfinity 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 SYZfibrations and JenkinStrebel differentials.

14:30, Room T2233
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 pnilpotent subalgebras of a fixed dimension r for a pLie algebra g. These varieties are close relatives of the much studied class of varieties of rtuples of commuting pnilpotent matrices which remain highly mysterious when r>2. In this talk, I¹ll present some of the representationtheoretic 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 pnilpotent subalgebra of g for g a Lie algebra of a reductive algebraic group. This is joint work with J. Stark.

16:00, Room T2233
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 psubgroups of G. We investigate the relative stable category stmod_H(kG) of finitely generated modules modulo Hprojective modules. Triangles in this category correspond to Hsplit 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 V5227
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 subtriangulated 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 V2200
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 LannesSchwartz artinian conjecture and Stembridge's conjecture on stability of Kronecker coefficients of the symmetric group.
Friday, 06 February 2015

14:00, Lecture Hall XE0202
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 KhovanovRozansky knot invariant in terms of Hochschild homology of Soergel bimodules.

15:15, Lecture Hall XE0202
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 quasicoherent 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 tensorcategorical analogs of affine schemes, projective schemes, tangent bundles and fiber products.
Friday, 30 January 2015

14:00, Lecture Hall XE0202
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 wellchosen 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 quasihereditary structure might be a useful tool to understand the internal tensor product in more detail.

15:15, Lecture Hall XE0202
Simone Virili (Padova): Length functions on Grothendieck categories with applications to infinite group representations
Abstract: Let C be a Grothendieck category. A nonnegative 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 subobjects.
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 XE0202
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 nontrivial 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 XE0202
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 XE0202
Wassilij Gnedin (Cologne): HarishChandra modules of SL(2,R)
Abstract: At the ICM 1970 Gelfand posed the problem to classify the indecomposable finitedimensional representations of a certain quiver with relations and oriented cycles. Gelfands problem was motivated by the study of HarishChandra 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 JordanHöldermultiplicities, top and socle) as well as their AuslanderReiten translations in the derived category and their contragredient duals.
This talk is based on joint work with Igor Burban.

15:15, Lecture Hall XE0202
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 quasiprojective 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 prelambda structures on the coefficient theory of a point, which is the Grothendieck group of a Karoubian Qlinear tensor category.
This is joint work with Laurentiu Maxim.
Friday, 05 December 2014

14:00, Lecture Hall XE0202
ZhiWei 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 XE0202
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 2support 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
Friday, 14 November 2014
Thursday, 13 November 2014
Friday, 07 November 2014

13:15, Lecture Hall XE0202
Alexander Soibelman (Bonn): Quiver representations, parabolic connections, and the DeligneSimpson problem
Abstract: The additive and multiplicative formulations of the DeligneSimpson 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 CrawleyBoevey'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 XE0202 (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 taumultiplicity, is expressed in terms of the AuslanderReiten 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 nontrivial taumultiplicity occur naturally. As a special case we study the noncommutative real elliptic curves and present a Witt surface which is a noncommutative FourierMukai partner of the Klein bottle.

15:45, Lecture Hall XE0202 (45 minutes)
Dirk Kussin (Paderborn): Noncommutative smooth projective curves (Part II)
Abstract: See part I.
Friday, 31 October 2014

13:15, Lecture Hall XE0202
Ryo Kanda (Nagoya): Atommolecule 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 twosided 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 XE0202
Gustavo Jasso (Bonn): nabelian categories
Abstract: In this talk I will introduce nabelian categories and discuss their basic properties. These categories, which are higher analogs of abelian categories, arise naturally in the study of higher AuslanderReiten theory. If time permits, I will describe a connection with GeissKellerOppermann's (n+2)angulated categories.

16:00, Lecture Hall XE0202
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 3CalabiYau 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 XE0202
Chao Zhang (Bielefeld): BrauerThrall 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 BrauerThrall type theorems for the bounded derived category of a finitedimensional algebra over an algebraically closed field are obtained. The first BrauerThrall type theorem says that derived bounded algebras are just derived finite algebras. The second BrauerThrall 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 XE0202
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 TamarkinTsygan 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_infinityalgebras 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 V2210/216
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 PoissonLie 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 wellknown 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 differentialgraded 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 AuslanderReiten quivers, allowing us to identify Coxeter functors as some kind of spectral Nakayama functors.

14:30, Lecture Hall H8
XiaoWu 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 homspaces 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 Extfunctor is a consequence of the HiltonRees 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
Friday, 13 June 2014
Thursday, 12 June 2014
Friday, 06 June 2014

13:15, Lecture Hall H8
Ivo Dell'Ambrogio (Lille): GrothendieckNeeman duality and Wirthmüller isomorphisms
Abstract: In this talk I will review Neeman's proof of Grothendieck duality in algebraic geometry and FauskHuMay'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 coproductpreserving exact tensor functor between two rigidlycompactly 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): Rmatrices, cellularity, and tilting modules at roots of unity
Abstract: Rmatrices 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 TemperleyLieb 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: SchurWeyl, skew Howe, Koszul
Abstract: The talk relates classical dualities, like SchurWeyl 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 XE0202
Colin Ingalls (Fredericton): Rationality of the BrauerSeveri 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 BrauerSeveri variety of a Sklyanin algebra is rational.
Friday, 16 May 2014

Colloquium in honour of the 75th birthday of Helmut Lenzing
10:00, Room V2210/216
Osamu Iyama (Nagoya): CohenMacaulay representations of GeigleLenzing 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 GeigleLenzing complete intersections. We will study Lgraded CohenMacaulay Rmodules, and show that there always exists a tilting object in the stable category. As an application we study when (R,L) is drepresentation finite in the sense of higher dimensional AuslanderReiten theory. This is a joint work with Herschend, Minamoto and Oppermann.

Colloquium in honour of the 75th birthday of Helmut Lenzing
11:30, Room V2210/216
Steffen Oppermann (Trondheim): Recollements for GeigleLenzing weighted projective varieties
Abstract: In my talk I will discuss GeigleLenzing 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 GeigleLenzing 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 GLprojective 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 GeigleLenzing type weights also on hypersurfaces in other projective varieties.

Colloquium in honour of the 75th birthday of Helmut Lenzing
14:00, Room V2210/216
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 GromovWitten 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 V2210/216
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 AuslanderReiten translation.

Colloquium in honour of the 75th birthday of Helmut Lenzing
16:45, Room V2210/216
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 V2210/216
RagnarOlaf 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 V2210
RagnarOlaf Buchweitz (Toronto): Higher RepresentationInfinite Algebras from Geometric Tilting Objects
Abstract: We will report on joint work with Lutz Hille on the recent notion of higher representationinfinite 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 representationinfinite 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 sheaflike.
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 HillePerling that comprises the classical notion of a helix, gives rise to such pairs of drepresentationinfinite algebras and their accompanying higher (d+1)preprojective algebras, thereby providing plenty of examples.
Intriguingly, such algebras also arise on nonFano varieties, such as the second Hirzebruch surface or some nonisolated 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 representationinfinite algebras are precisely the ones arising as endomorphism rings of minimal global dimension of sheaflike 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 algebraicallyclosed field can be realised as a Grassmannian of submodules for the Beilinson algebra. Using that every Grassmannian has a natural (in general nonreduced) 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
ZhiWei 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 nonpointed 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 quasicoherent modules on a scheme, in the differentialgraded 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 HFplocal 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 selfcontained 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 bicommutator rings and derived completion
Abstract: I will discuss (universal) properties of derived bicommutator 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 BarrettPriddyQuillen, the stable homotopy groups (of spheres) are isomorphic to the Kgroups of the category of finite pointed sets regarded as a Waldhausen category. The objective of my work is to approach the calculation of these Kgroups in a way analogous to the work of Grayson who has calculated the Kgroups 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 ARtranslation in finite type cluster categories, and analogues
Abstract: A cluster category of Dynkin type admits a finite cyclic action given by the AuslanderReiten translation, which induces an action on the set of clustertilting 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 "nonnesting partitions", which are equinumerous with the clustertilting objects. There is an analogous cylic action on the nonnesting partitions. Conjecturally, there are natural bijections between the two sets which intertwine these two actions, but the nonnesting side of the story is not well understood. In particular, a useful representationtheoretic understanding of the nonnesting 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 V2210/216
Julia Sauter (Bielefeld): The Auslander bijections: Introduction and examples
Abstract: This talk is an introduction of Auslander’s old approach to dissect modA for an artin algebra A with lattices [> Y) and C[> Y) of equivalence classes of morphisms ending in a fixed Amodule 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 finitedimensional Kalgebra over an algebraically closed field K, then C[> Y ) can be identified with (a union of) quiver Grassmannians.

11:15, Room V2210/216
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 V2210/216
Hugh Thomas (New Brunswick): Cofinite quotientclosed 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 kQmod is cofinite if it is full and contains all but finitely many of the indecomposables. We show that the cofinite quotientclosed subcategories of kQmod are naturally in bijection with the elements of the Weyl group associated to Q. We also extend this result to hereditary algebras which are finitedimensional 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 V2210/216
Osamu Iyama (Nagoya): Stable categories of onedimensional hypersurface singularities
Abstract: We show that the stable categories of graded CohenMacaulay modules over onedimensional hypersurface singularities k[x,y]/(f) with standard grading have tilting objects U. Moreover we show that their endomorphism algebras are 2representation 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 V2210/216
Idun Reiten (Trondheim): Some aspects of tautilting theory
Abstract: The theory of tautilting 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 tautilting 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 Ktheory. By a result of Grayson, this product generalizes the usual intersection product on a nonsingular algebraic variety.
Friday, 24 January 2014

13:15, Room V2210
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 Rmodules, 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 Rlinear 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 V2210
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 FourierMukai transformations to the world of dg algebras.
Friday, 17 January 2014

14:15, Lecture Hall H10
Julia Sauter (Leeds): Type ASpringer 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 KLRalgebras. 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_nequioriented quiver case I found a cell decomposition of the quiver flag varieties which is a geometric realization of a cellular algebra structure of this KLRalgebra. 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): Noncommutative Poisson structures for quasitriangular Hopf algebras
Abstract: Let A be an associative and unital algebra over a commutative ring. By definition, a noncommutative 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 degree2element inside the cohomology ring of a quasitriangular Hopf algebra H gives rise to a noncommutative 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 LabardiniFragoso associated a quiver with potential (Q, W). By Amiot's work, there is a 2CalabiYau 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 nonempty 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 CalabiYau reduction as cutting and the AuslanderReiten 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 V5227
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
Thursday, 12 December 2013

Faculty Colloquium
17:15, Room V2210/216
Markus Reineke (Wuppertal): Wallcrossing formulas
Abstract: Wallcrossing formulas arose in the last few years in string theory and algebraic geometry, most notably in the work of Kontsevich and Soibelman on motivic DonaldsonThomas invariants. The aim of the talk is to give a nontechnical introduction to the ideas leading to wallcrossing 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 V3201
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 V2210
Philipp Lampe (Bielefeld): Divisor class groups of cluster algebras
Abstract: We wish to present a necessary and sufficient and computercheckable 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 V2210
Andrew Hubery (Bielefeld): Subgroups of pointed Weyl groups, and noncrossing partitions
Abstract: Starting from a lattice with a nondegenerate bilinear from, and a complete exceptional sequence, we can associate a Weyl group and a Coxeter element, and hence a poset of noncrossing 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 noncrossing partitions. We therefore obtain interesting monomorphisms of Weyl groups where the image is in general not a parabolic subgroup.

16:00, Room V2210
Christof Geiss (Mexico City): On the uniqueness of nondegenerate potentials for mutation finite quivers
Abstract: Recall that by a result of FeliksonShapiroTumarkin 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 (DerksenOwen);
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 nondegenerate potential.
Remarks: (1) In the cases C0), C1) and C4) there are definitively several (weak) right equivalence classes of nondegenerate potentials. The main open case is C2).
(2) Nonuniqueness of nondegenerate 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. LabardiniFragoso 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 kalgebra. Let M be a linear kmodule and let Γ be the extalgebra 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 Hmod without nonzero Extprojective modules. This answers a question of AdamChristiaan von Roosmalen.
Friday, 15 November 2013

14:15, Room V2210/216
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 XiaoWu Chen and Dong Yang.
Saturday, 09 November 2013
Friday, 08 November 2013
Thursday, 07 November 2013

Workshop "Polynomial Functors and Schur Algebras"
10:00, Room V2210/216

Faculty Colloquium
17:15, Room V2210/216
Catharina Stroppel (Bonn): From classical SchurWeyl duality to quantized skew Howe dualities
Abstract: Classical SchurWeyl 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 U2113
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 CohenMacaulay modules of covariants for the action of SL(n) on the set of m nvectors. It also allows to describe the equivariant simple Dmodules, 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 U2113
Yuriy Drozd (Kiev): Tilting and resolutions for singular curves
Abstract: For singular projective curves we construct a full embedding of the derived category of quasicoherent 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 noncommutative curves. (This is a joint work with I.Burban and V.Gavran.)
Friday, 18 October 2013

14:00, Room U2113
Greg Stevenson (Bielefeld): Graded modules versus dgmodules
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 dgalgebra with trivial differential and form the derived category of dgmodules. 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 dgmodules.

15:15, Room U2113
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 V3204
Jeanne Scott (Chennai): The twist, by matchings
Abstract: The Grassmannian 'twist' is a birational automorphism of the typeA 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 semisimple 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 V3204
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 representationtheoretic and homological aspects. This is joint work with Edward L. Green.

14:30, Lecture Hall H9
Ryan Kinser (Boston): Type A quiver loci and KazhdanLusztig varieties
Abstract: We show how to embed a representation variety of a type A quiver into a KazhdanLusztig 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 CohenMacaulay, and also leads to formulas for cohomology and Kclasses of the orbit closures.

16:00, Lecture Hall H9
Grzegorz Bobinski (Torun): Moduli spaces for quasitilted 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 quasitilted algebras always have a singular moduli space. During my talk I will discuss the proof of the other implication in the case of quasitilted algebras.
Friday, 19 July 2013
Friday, 21 June 2013

13:15, Room V2213
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 LabardiniFragoso 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 finitedimensional, 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 finitedimensional symmetric Jacobian algebras. Moreover, these Jacobian algebras are closely related to Brauer graph algebras arising naturally from triangulations of the surface.

14:30, Room V2213
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 kcategories using Grothendieck constructions of colax functors from a small category I to the 2category of kcategories. 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 kcategories and bimodules over them to recover triangular matrix algebras (or more generally tensor algebras of kspecies).
Friday, 14 June 2013
Saturday, 08 June 2013
Friday, 07 June 2013

13:15, Room V2213
AdamChristiaan 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 2CalabiYau 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 infinitygon.

14:30, Room V2213
Julia Worch (Kiel): Module categories and AuslanderReiten 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 AuslanderReiten theory and thereby obtain information concerning the occurrence of the corresponding modules within the AuslanderReiten quiver of the Beilinson algebra.
Friday, 31 May 2013

13:15, Room V2213
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 quivergraded 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 quivergraded Springer theory.

14:30, Room V2213
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 V2213
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 V2213
Julia Sauter (Leeds): Classical and quivergraded 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 BBDdecomposition theorem gives a parametrization of (graded) indecomposable projectives and simple modules for the Steinberg algebras. The two main examples are classical and quivergraded 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 quivergraded Springer theory are quiver Hecke algebras (=KLRalgebras) introduced by KhovanovLauda and Rouquier. My result here is an explicit calculation of Steinberg algebras in a more general setup.

15:15, Room V2213
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 V2213
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 Yonedalemma 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 V2213
Phillip Linke (Bielefeld): Computational approach to the Artinian conjecture (Part II)
Abstract: See Part I.

16:00, Room V2213
Sven Meinhardt (Wuppertal): Motivic DTinvariants 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 3folds. After that, an alternative noncommutative resolution using quivers with potential is given. Finally, I will briefly introduce DonaldsonThomas 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 V2213
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 AvellaAlaminos & Geiß.
Friday, 12 April 2013

Workshop Noncrossing Partitions
09:00, Room V2210/216
Claus Michael Ringel (Bielefeld): The noncrossing 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 noncrossing partitions in terms of the representation category of a Dynkin quiver: they exhibited, for example, a bijection between the noncrossing 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 nonsimply 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 wellknown in the cases A_n, but seemed to be quite mysterious in the remaining cases.

Workshop Noncrossing Partitions
10:15, Room V2210/216
Friedrich Götze (Bielefeld): Free Probability and Noncrossing Partitions

Workshop Noncrossing Partitions
11:30, Room V2210/216
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 V2210/216
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, Coxetersortable elements and noncrossing partitions.

Workshop Noncrossing Partitions
15:15, Room V2210/216
Henning Krause (Bielefeld): A Ktheoretic 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 Extspaces). 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 V2210/216
Gennadiy Chistyakov (Bielefeld): Distributions of commutators and anticommutators
Thursday, 11 April 2013

Workshop Noncrossing Partitions
15:30, Room V2210/216
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 selfcontained; 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 socalled local subgroups (e.g., the normalisers of certain psubgroups).
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 LabardiniFragoso. We determine the representation type of (almost) all Jacobian algebras P(Q,S) arising from a 2acyclic quiver Q and a nondegenerate 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 rradical and rsocle 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 pgroup, there is universal function for computing the kernel bundles as modules over the structure sheaf of the Grassmannian of rplanes in n space. These ideas also extend to various sorts of subalgebra of restricted pLie algebras.
Friday, 18 January 2013

13:15, Lecture Hall H6
Lennart Galinat (Köln): Orlov's Equivalence and Maximal CohenMacaulay 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_{n1} 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 2functors to give an explicit construction of the Extalgebra of Weyl modules, which in particular yields a multiplicative (up to sign) basis.
Thursday, 17 January 2013

10:15, Room V5227
Vanessa Miemietz (Norwich): 2functors and homological duality
Abstract: We will explain how certain 2functors 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 Zmodule, 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 „wellbehaved“ 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 KrullGabriel 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 KrullGabriel dimension is bounded by one.

16:00, Lecture Hall H6
Dieter Vossieck (Bielefeld): Representationdiscrete algebras and the second BrauerThrall conjecture
Abstract: We want to discuss the following statement which is equivalent to the second BrauerThrall conjecture / theorem of NazarovaRoiter / theorem of BautistaBongartz:
Over an algebraically closed field k , let k[[Q]] be the complete pathalgebra 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 representationinfinite 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): Noncommutative 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
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 profinite 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 cohomology vanishing condition can not be reduced in general. Moreover, the Gorenstein projective conjecture is true for CMfinite 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 Xacyclic 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 (nonconstructive) proof that there exist objects in the derived category of graded modules over a certain graded nonNoetherian 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 (finitedimensional) Qrepresentation 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 $B_Q$ derivedequivalent 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 LabardiniFragoso (Bonn): On a family of species with potentials
Abstract: I will talk on work in progress, joint with Andrei Zelevinsky, regarding possible extensions of DerksenWeymanZelevinsky's mutation theory of quivers with potentials to the setting where the matrix encoded by the quiver is not skewsymmetric but rather skewsymmetrizable.
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 Rmodules. 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 onevariable 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 nvariable versions can be attacked using more complicated combinatorics of higherdimensional 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 LangeHegermann

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 Genclasses 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 pureinjective torsion part. We classify all large tilting sheaves which have a noncoherent 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 AngeleriHügel.
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 dualanalogous 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 dualanalogous to a quasicoherent 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 representationtheoretic 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 2CalabiYau triangulated categories
Abstract: For a CalabiYau triangulated category of CalabiYau dimension d, the decomposition of this category as triangulated subcategories is determined by the decomposition of any dcluster tilting subcategory. This induces that the quivers of cluster tilting objects in a 2CalabiYau triangulated category are connected or are not connected at the same time. As an application, the tstructures (or cotstructures) of an indecomposable 2CalabiYau 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 semidirect 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 DrinfeldJimbo 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): BatalinVilkovisky 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 SchoutenNijenhuis bracket, and the differential forms are what we call a BatalinVilkovisky 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 CalabiYau algebra is a BatalinVilkovisky algebra is extended to twisted CalabiYau 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 CrawleyBoevey (Leeds): Dmodules for nodal curves and multiplicative preprojective algebras
Abstract: In work on the DeligneSimpson 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 torsionfree sheaves on nodal curves. This talk is a response to that question. I will discuss various categories of Dmodules 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 tstructure: some examples
Abstract: Let A be a finitedimensional algebra. Then there is a oneto one correspondence between silting objects in K^b(proj A) and bounded tstructures 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, AuslanderReiten quivers, and cluster categories in finite types. This perspective leads to new combinatorial objects called multicluster complexes, of which I will discuss possible relations to certain identifications in the repetition quiver in terms of AuslanderReiten 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 (semiexact) 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 (MCMrepresentation finite) Gorenstein singularity: the singularity category of Buchweitz and the relative singularity category of a noncommutative (Auslander) resolution, which was studied in joint work with Burban. We show that these categories mutually determine each other in the case of ADEsingularities in any Krull dimension. Knörrer's periodicity theorem yields a wealth of nontrivial examples. This is joint work with Dong Yang.
Friday, 15 June 2012

13:15, Lecture Hall H10
Raquel Simoes (Leeds): Homconfigurations and noncrossing partitions
Abstract: We will give a classification of maximal Homfree sets of indecomposable objects (the Homconfigurations 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 representationfinite 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 Rouquierdimension which is the smallest such integer. In a recent paper by BallardFaveroKatzarkov, 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 pgroup 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 kEmodule called the generic kernel. In the case where M is a kEmodule 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): Cotstructures and costability
Abstract: In this talk we introduce the ideas of cotstructures and co stability conditions and compare and contrast with tstructures and stability conditions. We show that the space of costability 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 Rmodules is right determined by an Rmodule 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 nonzero projective direct summand. What we encounter is an intimate relationship to the vanishing of Ext^2.
Friday, 23 March 2012

13:15, Room U2205
Jan Schröer (Bonn): Generic CalderoChapoton functions and generalized clusters
Abstract: This is joint work with Giovanni Cerulli Irelli and Daniel LabardiniFragoso. To any algebra A defined as a factor of a completed path algebra we associate an algebra CC(A) generated by generalized CalderoChapoton 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 finitedimensional algebras.

14:30, Room U2205
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 Fisomorphism in mod p cohomology (i.e., the kernel is nilpotent and every element has some ppower 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 U2205
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 U2205
Jan Stovicek (Prague): Resolving subcategories for a commutative noetherian ring
Abstract: Given a commutative notherian ring R, we classify resolving subcategories of modR 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 U2205
Markus Perling (Bochum): Equivariant resolutions and maximal CohenMacaulay modules over affine toric varieties
Abstract: We present a combinatorial framework which allows to translate free resolutions of Z^ngraded modules over the polynomial ring in n variables into projective resolutions over certain incidence algebras. We present two applications: 1. we explicitly determine the Bettinumbers 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 V2213
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 2group 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 V2213
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 V2213
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 Rmodules 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 V2213
Irakli Patchkoria (Bonn): On the algebraic classification of module spectra
Abstract: For any Salgebra 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 BousfieldWolbert algebraic classification of isomorphism classes of objects in the derived category of R. In the case of global dimension two, the plocal real connective Ktheory, the first Johnson Wilson spectrum E(2) and the truncated BrownPeterson spectrum BP<1>, for an odd prime p, serve as our main examples. Examples of Salgebras 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 V2213
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 kQmodules. It seems to be worthwhile to consider also representations of Q over arbitrary finitedimensional kalgebras 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 Lambdamodules, where Lambda = kQ[epsilon] is the path algebra of Q over A. The algebra Lambda is a 1Gorenstein algebra, thus the torsionless Lambdamodules are known to be of special interest (as the Gorensteinprojective or maximal CohenMacaulay 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 kQmodules 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 Lapproximation. We also describe the embedding of the AuslanderReiten quiver of mod kQ into that of T.
This is a report on current joint investigations with Zhang Pu (SJTU).

16:00, Room V2213
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 1domestic 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 KrullGabriel dimension of any 1domestic 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 V2213
Artem Lopatin (Omsk): Matrix identities with forms
Abstract: A linear group GL(n) acts on dtuples 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), 389401] established generators of R(n,d) and Zubkov [Algebra and Logic 35 (1996), No. 4, 241254] 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 Tideal of identities of $M_n$ with forms is finitely based.

14:30, Room V2213
Grzegorz Bobinski (Torun): Semiinvariants of concealedcanonical algebras
Abstract: The descriptions of the algebras of semiinvariants for regular dimension vectors over Euclidean and canonical algebras are basically identical. Thus it is natural to expect that these descriptions generalize to arbitrary concealedcanonical algebras. The main result of my talk states that this is really the case.

16:00, Room V2213
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 1domestic 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 KrullGabriel dimension of any 1domestic 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 C0269
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 higherrank 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 padic integration and a Kirillov orbit method to parameterize representations by coadjoint orbits.
Friday, 18 November 2011

13:15, Room V2213
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 Clustertilting 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 V2213
Roger Wiegand (Lincoln): BrauerThrall theorems and conjectures for commutative local rings
Abstract: The BrauerThrall Conjectures, now theorems, were originally formulated in terms of representations of finitedimensional 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 CohenMacaulay modules over CohenMacaulay local rings.

16:00, Room V2213
Sylvia Wiegand (Lincoln): The Anatomy of a Stranger: A (Barely) nonNoetherian 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 nonNoetherian 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 threedimensional nearNoetherian 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 2generated. We also mention more mysterious generalizations to higher dimensions.
Friday, 04 November 2011

14:30, Room V2213
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 V2213
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 nonfunctoriality 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 V2213
AnnaLouise 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 Homfunctor 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 V2213
Ralf Meyer (Göttingen): Hereditary exact categories from equivariant bivariant Ktheory
Abstract: Together with Ryszard Nest and my students Rasmus Bentmann and Manuel Köhler, I have been studying Universal Coefficient Theorems in bivariant Ktheory for C*algebras. The idea there is to find a Ktheoretic 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 ADEquivers, 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 V2213
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 AvramovBuchweitz for complete intersection rings. The proof uses the support theory of Benson, Iyengar, and Krause.

14:30, Room V2213
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 V2213
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 V4119
Gil Kaplan (Tel Aviv): Some new results on finite Tgroups
Abstract: A group is called a Tgroup 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 Tgroups. We give new results in this subject and new characterizations of solvable Tgroups. 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 pgroups, 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 ModA for an hereditary algebra are described with the notion of universal localizations. In this talk an alternative approach via QcohX 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 kCat the 2category of kcategories. An (op)lax functor from I to kCat can be regarded as a generalization of a lax group action on a kcategory. 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 Gsets, and Galois morphisms of Gsets, respectively, and show how the properties of the usual Burnside ring can be extended to these new rings.
Friday, 22 July 2011
Thursday, 21 July 2011

10:30, Lecture Hall H8
Helene Tyler (Riverdale): AuslanderReiten Layers in the Rhombic Picture
Abstract: The GabrielRoiter measure was first introduced by Roiter in his 1968 proof of the first BrauerThrall conjecture. For a finite length module, the pair consisting of the GR measure and the GRcomeasure 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 AuslanderReiten 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_nequioriented quiver flag varieties
Abstract: We study quiver flag varieties for the A_nequioriented 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_nequioriented 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, CharbonneauJodoin and Plamondon. They can be seen as finitedimensional 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 LabardiniFragoso.
We will present a complete derived equivalence classification of these algebras, thereby generalizing the classifications of clustertilted 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 AvellaAlaminos and Geiss for gentle algebras and computed recently for the algebras in question by DavidRoesler 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
RagnarOlaf 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
RagnarOlaf 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 CohenMacaulay modules, equivariant for a suitable group action.
We will explain how GrothendieckSerre 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 inclusionpreserving bijection between, on the one hand, the twistclosed 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 nonaffine schemes.

16:00, Lecture Hall H10
Mike Prest (Manchester): Superdecomposable pureinjectives over tubular algebras
Abstract: A module over an artin algebra is pureinjective (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) pureinjectives 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 pureinjective 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 CohenMacaulay 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 CohenMacaulay 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 CohenMacaulay 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 columnfinite 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 CrawleyBoevey and Ringel in the early 90's. In the late 90's, Biane, BradyWatt, 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 IgusaSchiffler. 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 $n(n1)/2$ (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 $n(n1)/2$ and we can determine the structure of the subgroup generated by these matrices.
Friday, 03 June 2011

13:15, Lecture Hall H4
ZhiWei Li (Bielefeld): Derived minimal dg algebras
Abstract: We call a dg algebra derived minimal, if its derived category of dgmodules 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 kGmodules for the Klein four group.

14:30, Lecture Hall H4
Shengkui Ye (Singapore): An application of representation theory to the computation of algebraic Kgroups of group rings
Abstract: In this talk, an injection of homology of groups into algebraic Kgroups 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 Ktheory and so on.

16:00, Lecture Hall H4
Chris Brav (Hannover): Pingpong and exceptional vector bundles
Abstract: We present a strategy for proving that full exceptional collections of vector bundles on projective nspace 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 pingpong lemma of FrickeKlein to solve this problem in low dimensions, thus providing a new proof of constructibility of exceptional collections in some cases. We expect a similar pingpong argument to give constructibility on projective nspace 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 TateVogel 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 HirzebruchRiemannRoch 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 Gspace 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 kGmodule 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 Gposet 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 Gposet, 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 SnashallSolberg, based on Hochschild cohomology rings, works perfectly for such a block algebra. Furthermore the SnashallSolberg 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 noncommutative geometry is to refrain from passing to the quotient and construct a noncommutative algebra that encodes some of the data lost in passing to X/~. In this example (see Connes book for details) the relevant noncommutative 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 noncommutative curve. This is similar to treating the preprojective algebra of a wild hereditary algebra as the homogeneous coordinate ring of a noncommutative curve. The category of quasi coherent sheaves on this noncommutative 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 nonvanishing 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
Saturday, 07 May 2011
Friday, 06 May 2011
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 nonaffine matrix factorizations
Abstract: We introduce a category of nonaffine "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 PolishchukVaintrob. We show how these nonaffine 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 PolishchukVaintrob from hypersurfaces to complete intersections of arbitrary codimension. If time permits we also show how this construction allows generalizations of known results of AvramovBuchweitz and AvramovIyengar 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 3CYvarieties
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 3CYvariety. 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_4subalgebras
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 mid90s they were described in terms of quivers and relations by W. CrawleyBoevey and R. VilaFreyer. The path algebra of D_4 appears to be the minimal nonbiserial 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 1quasihereditary algebras
Abstract: The topic of this talk is a subclass of quasihereditary basic algebras, called 1quasiherediditary (roughly speaking, any "possible" multiplicity in a JordanHölderfiltration 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 selfinjective algebras. If the Ringeldual of a 1quasihereditary algebra is also 1quasihereditary, 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 BazzoniStovicek, BraunGöbel, 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): KhovanovRozansky link homology via maximal CohenMacaulay approximations and the combinatorics of Soergel bimodules
Abstract: In this talk, I will outline a new construction of KhovanovRozansky 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 CohenMacaulay modules over the associated hypersurface and see what the KRconstruction looks like on this side. It will turn out that the modules we get can be expressed as maximal CohenMacaulay approximations to certain Soergel bimodules. The wellunderstood 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 Rmodules 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 PoincaréBirkhoffWitt 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 FominZelevinsky'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 GeißLeclercSchröer's nonquantized 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 CalabiYau structure on matrix factorization categories.
Sunday, 12 December 2010
Saturday, 11 December 2010
Friday, 10 December 2010
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
AdamChristiaan 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 AuslanderReitenTheory 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 semidefinite 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 Homfinite and representationdiscrete.
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 (ClujNapoca): Crossed products, Brauer groups and Clifford classes
Abstract: We introduce an equivalence between central simple strongly Ggraded algebras. Such classes are associated in a natural way to absolutely irreducible characters of semisimple Ggraded algebras. We study invariants of this equivalence relation, and also the structure of certain representatives of the equivalence classes. We show that Ggraded 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
Friday, 05 November 2010
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 (BuenosAires): 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.7190], adapting methods from [M. Farinati, A. Solotar, M. SuarezAlvarez, Hochschild homology and cohomology of generalized Weyl algebras. Ann. Inst. Fourier (Grenoble) 53(2), (2003) pp.465488].
Examples of such algebras are the quantum nth 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 SuarezAlvarez 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
Friday, 08 October 2010
Friday, 24 September 2010

10:00, Room V2216
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 kCat the 2category of klinear categories. As a generalization of a group action on a kcategory, we consider a functor X: I > kCat (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 > kCat.
The Morita type theorem characterizing derived equivalences by Rickard and Keller will be generalized in this setting.
Friday, 03 September 2010

13:15, Room V2216
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 V2216
Xiaojin Zhang (Nanjing): The existence of maximal orthogonal subcategories
Abstract: We build a connection between Auslander's 1Gorenstein algebras of global dimension 2 admitting trivial maximal 1orthogonal subcategories and tilted algebras of finite type, and we draw the quivers of such algebras. Moreover, all (n1)Auslander algebras admitting trivial maximal (n1)orthogonal subcategories can be determined.
Friday, 23 July 2010

13:15, Room T2220
Guodong Zhou (Bielefeld): The BatalinVilkovisky 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 BatalinVilkovisky structure, that is, there is a $\Delta$ operator over Hochschild cohomology which together with the cupproduct 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 T2220
Ivo Dell'Ambrogio (Singapore): On support varieties for (G) C*algebras
Abstract: Topological Ktheory, and its bivariant version KKtheory, are probably the most useful invariants of C*algebras and GC*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 wellbehaved theory of support varieties for (sufficiently nice) GC*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 T2220
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 V5227
Andrew Hubery (Leeds): Tangent Spaces, Massey Products, and Applications to Representation Theory
Abstract: We look again at an article by CrawleyBoevey 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 T2220
Julia Sauter (Leeds): On quiver flag varieties
Abstract: We review the quivergraded 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 HappelVossieck list.

14:30, Room T2220
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 T2220
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 CalderoChapotonSchiffler, BuanVatne, AssemBruestleCharbonneauJodoinPlamondon, 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 AvellaAlaminos 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 BrennerButler tilts.
Friday, 09 July 2010

13:15, Room T2220
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 BrennerButler 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 FominZelevinsky quiver mutation, and present applications to endomorphism algebras of clustertilting objects in 2CalabiYau categories as well as algebras of global dimension two.

14:30, Room T2220
Dan Zacharia (Syracuse): On AuslanderReiten components for selfinjective algebras I
Abstract: see Kerner

16:00, Room T2220
Otto Kerner (Düsseldorf): On AuslanderReiten 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 AuslanderReiten 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 T2220
Claudia Köhler (Bielefeld / Paderborn): Thick subcategories of selfinjective algebras
Abstract: The stable module category of a selfinjective 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 selfinjective standard algebra of finite representation type. There is an elaborate classification of these algebras by Riedtmann assigning to each selfinjective representationfinite 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 extensionclosed 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 selfinjective algebras.

16:00, Room T2220
AdamChristiaan 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 T2220
XiaoWu 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 T2220
Janine Bastian (Hannover): Derived equivalences for clustertilted algebras of Dynkin type D
Abstract: This is a joint work with Thorsten Holm and Sefi Ladkani.
Clustertilted algebras arise as endomorphism algebras of clustertilting 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 clustertilted algebra of Dynkin type is uniquely determined by its quiver.
In this talk we give a complete good mutation equivalence classification of clustertilted 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 T2220
Steffen Oppermann (Köln): Stable categories of higher preprojective algebras
Abstract: In this talk I will introduce the notion of nrepresentation finiteness, generalizing hereditary representation finite algebras. Inspired by the hereditary case (n=1) I will introduce higher preprojective algebras, and show that for nrepresentation finite algebras they are finite dimensional and selfinjective.
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)CalabiYau. It will be indicated how they can be identified with the (n+1)Amiot cluster category of the stable nAuslander algebras of the nrepresentation finite algebras.
Finally I will point out that the above results still hold if the assumption of nrepresentation 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 selfinjective, 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 CohenMacaulay modules.
Friday, 04 June 2010

Reading Course Cluster Algebras
13:15, Room T2220
Reiner Hermann (Bielefeld): Acyclic CalabiYau categories

Reading Course Cluster Algebras
14:30, Room T2220
XiaoWu Chen (Bielefeld / Paderborn): Cluster categories of canonical algebras

16:00, Room T2220
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 T2220
Phillip Linke (Bielefeld): Mutation in triangulated categories

Reading Course Cluster Algebras
14:30, Room T2220
Claudia Köhler (Paderborn / Bielefeld): Triangulated orbit categories

16:00, Room T2220
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 T2220
ZhiWei Li (Bielefeld): Clustertilting in a general framework

Reading Course Cluster Algebras
14:30, Room T2220
Zhe Han (Bielefeld): From triangulated categories to cluster algebras

16:00, Room T2220
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 Fpolynomials and gvectors for cluster algebras with arbitrary skewsymmetric exchange matrices.
Friday, 23 April 2010

Reading Course Cluster Algebras
13:15, Room T2220
Guodong Zhou (Paderborn): Cluster categories

Reading Course Cluster Algebras
14:30, Room T2220
Jue Le (Paderborn): Clustertilted algebras

15:45, Room T2220
Matthias Künzer (Aachen/Stuttgart): From ntriangles to Heller triangulated categories (after S. Thomas)
Abstract: In Heller triangulated categories, 2triangles are just Verdier triangles, whereas 3triangles are particular Verdier octahedra. More generally, there are ntriangles which enjoy the same properties as Verdier triangles: one can form cones, rotate, prolongate morphisms (nonuniquely). These ntriangles form a simplicial set. The usual way to specify these ntriangles is: define npretriangles by exactness properties, pick an isomorphism theta_n between induced shift functors and let an ntriangle be an npretriangle X such that X theta_n = id. Sebastian Thomas discovered that this process can be reversed: picking a subset {ntriangles} in {npretriangles} 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 {ntriangles} 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 T2220
Changchang Xi (Beijing): Derived equivalences and cohomological approximations
Abstract: In this talk, we shall provide an abstract method to construct derived equivalences between \PhiAuslanderYoneda algebras from an almost Dsplit 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, CalabiYau categories and Frobenius categories. This is a part of my joint work with Hu and Koenig.
Friday, 16 April 2010