Seminar
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.
For a regular email announcement please contact birep.
Future Talks
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 Our construction relies on unpublished work by Darpö and Iyama.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, 04 November 2016

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.

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, 18 November 2016

Room U2135
Sondre Kvamme (Bonn): tba

Room U2135
Rosanna Laking (Bonn): tba
Friday, 25 November 2016

Room U2135
Matthew Pressland (Bonn): tba
Friday, 02 December 2016

Room U2135
Peter Patzt (Berlin): tba

Room U2135
Andrew Hubery (Bielefeld): tba
Seminar Archive
Friday, 26 August 2016

14:15, Room V5227
ZhiWei Li (Xuzhou): A homotopy theory of additive category with suspensions
Abstract: We give a definition of partial onesided triangulated categories. We show that complete cotorsion pairs in exact categories, torsion pairs and mutation pairs in triangulated categories all extend to partial onesided triangulated categories. We prove that partial onesided triangulated categories yield onesided riangulated categories by passing to stable categories. We give three areas of application of this result. The first one is the constructions of stable abelian and exact categories which extend work of KoenigZhu, KellerReiten and KussinLenzingMeltzer. The second one is the construction of stable triangulated categories which allows us to model IyamaYoshino subfactors of triangulated categories via Quillen closed model structures. The last one is to develop a homotopy theory of additive categories with suspensions via GabrielZisman localization which leads to a Buchweitz type theorem in triangulated categories. This theorem extends the recent work of Wei and IyamaYang which are generalizations of Buchweitz's work on singularity categories. As a corollary we give a triangle equivalence between Verdier quotients and IyamaYoshino subfactors of triangulated categories under suitable conditions.
Thursday, 21 July 2016

16:15, Room V3204
Anna Felikson (Durham): Geometric realizations of quiver mutations
Abstract: Mutations of quivers are simple combinatorial transformations introduced in the context of cluster algebras, they appear (sometimes completely unexpectedly) in various domains of mathematics and physics. In this talk we discuss connections of quiver mutations with reflection groups acting on vector spaces and with groups generated by point symmetries in the hyperbolic plane. We show that any mutation class of rank 3 quivers admits a geometric presentation via such a group and that the properties of this presentation are controlled by the Markov constant p^2+q^2+r^2pqr, where p,q,r are the weights of the arrows in the quiver. This is a joint work with Pavel Tumarkin.
Friday, 15 July 2016

13:15, Room C01142
Ann Kiefer (Bielefeld): Units in Integral Group Rings via Fundamental Domains and Hyperbolic Geometry
Abstract: The motivation of this work is the investigation on the unit group of an integral group ring U(ZG) for a finite group G. By the WedderburnArtin Theorem, the study of U(ZG) may be reduced, up to commensurability, to the study of SL_n(O) for n ≥ 1 and O an order in some division ring D. There exists descriptions of a finite set of generators for a subgroup of finite index in SL_n(O) for a large number of cases. Excluded from this result are the socalled exceptional components of QG.
Our work consists in finding a presentation, for SL_n(O) associated to some of these exceptional components. In all the cases we treat, the group SL_n(O) has a discontinuous action on hyperbolic space of dimension 2 or 3, on hyperbolic space of higher dimensions, or on some product of hyperbolic spaces. By constructing fundamental domains for these discontinuous actions, we get generators for the groups in question.

14:30, Room C01142
Martin Kalck (Edinburgh): Knörrertype equivalences for twodimensional cyclic quotient singularities
Abstract: We construct triangle equivalences between singularity categories of twodimensional cyclic quotient singularities and singularity categories of a new class of finite dimensional algebras, which we call Knörrer invariant algebras. In the Gorenstein case, we recover a special case of Knörrer’s equivalence for hypersurfaces. Time permitting, we’ll explain how this led us to a formula for the Ringel duals of certain strongly quasihereditary algebras. This is based on joint work with Joe Karmazyn.

16:00, Room C01142
Grzegorz Bobiński (Torun): On nonsingularity in codimension one of irreducible components of module varieties over quasitilted algebras
Abstract: For a given dimension vector over a triangular algebra the closure of the set of modules of projective dimension at most 1 is an irreducible component (if nonempty). There are results showing that this component should have good geometric properties. For example, if the dimension vector is the dimension vector of a directing (non necessarily indecomposable) module, then this component is nonsingular in codimension one. A new result (joint with Zwara) says that the same holds for the dimension vectors of regular modules over concealed canonical algebras. We hope to generalize these results to arbitrary dimension vectors over quasitilted algebras.
Friday, 01 July 2016

13:15, Room C01142
Hagen Meltzer (Szczecin): Exceptional objects for nilpotent operators with invariant subspace
Abstract: This is a report on joint work with Piotr Dowbor (Torun) and Markus Schmidmeier (Boca Raton). We study (graded) vector spaces equipped with a nilpotent operator of nilpotency degree n and an invariant subspace. This problem is related to an old one stated by Birkhoff and recent results were obtained by RingelSchmidmeier, by Simson and in joint work with Kussin and Lenzing investigating stable vector bundle categories for weighted projective lines. In particular for n=6 the category is of tubular type.
We study exceptional objects in this category and show that each of them can be exhibited by matrices having as coefficients only 0 and 1.

14:30, Room C01142
Alexander Kleshchev (Eugene): Stratifications of KhovanovLaudaRouquier algebras
Abstract: We review standard module theory for KhovanovLaudaRouquier algebras of finite and affine types, its connections with PBW bases in quantum groups, and affine highest weight categories. Time permitting, we give an applications to blocks of symmetric groups and Hecke algebras.
Friday, 24 June 2016

14:15, Room C01142
Alexander Merkurjev (Los Angeles): Rationality problem for classifying spaces of algebraic groups
Abstract: Many classical objects in algebra such as simple algebras, quadratic forms, algebras with involutions, Cayley algebras etc, have large automorphism groups and can be studied by means of algebraic group theory. For example, for each type of algebraic objects there is an algebraic variety (called the classifying space of the corresponding algebraic group) that classifies the objects. The simpler the structure of this variety, the simpler the classification. For example, rationality of the classifying variety means that the objects can be described by algebraically independent parameters. I will discuss the rationality property of classifying varieties.
Friday, 17 June 2016

13:15, Room C01142
George Dimitrov (Bonn): Unstable exceptional objects in hereditary categories
Abstract: On the way of describing the entire Bridgeland stability spaces on some quivers we handled unstable exceptional objects in hereditary categories, whereby specific pairwise relations between exceptional objects were utilized. In this talk I will tell more about this.

14:30, Room C01142
Xin Fang (Cologne): On degenerations of flag varieties
Abstract: Motivated by the PBW filtration of Lie algebras, E. Feigin defined the degenerate flag varieties, which are flat degenerations of the corresponding flag varieties. The purpose of this talk is to introduce a new family of (flat) degenerations of complete flag varieties of type A, called linear degenerate flag varieties, by classifying flat degenerations of a particular quiver Grassmannian. The geometry of these degenerations will also be presented. This talk is based on a joint work with G. Cerulli Irelli, E. Feigin, G. Fourier and M. Reineke.

16:00, Room C01142
Fritz Hörmann (Freiburg): Fibered multiderivators, (co)homological descent and Grothendieck's six operations
Abstract: The theory of derivators enhances and simplifies the theory of triangulated categories. We propose a notion of fibered (multi)derivator, which similarly enhances fibrations of (monoidal) triangulated categories. We present a theory of cohomological as well as homological descent in this language. The key is a generalization of the notion of ``fundamental localizer'' to diagrams in a category with Grothendieck topology. The main motivation is a descent theory for Grothendieck's six operations. We will also explain how a (classical) six functor context can be defined as a fibered multicategory, thus giving a simple precise definition including all possible compatibility relations between the six functors.
Friday, 10 June 2016

14:15, Room C01142
Moritz Groth (Bonn): Characterizations of abstract stable homotopy theories
Abstract: The typical triangulated categories arising in nature are homotopy categories of suitable stable homotopy theories in the background. This applies to derived categories of abelian categories as well as to the stable homotopy category of spectra. In this talk we discuss various characterizations of abstract stable homotopy theories, thereby describing aspects of the calculus of chain complexes. Moreover, each of these characterizations specializes to an answer to the following question: what is the defining feature of the passage from (pointed) topological spaces to spectra?
Friday, 03 June 2016

14:15, Room C01142
William Sanders (Trondheim): A Pointless approach to triangulated categories
Abstract: In the past several decades, algebraists have used various notions of support to study the thick subcategories of certain triangulated categories. However, each of these notions require the triangulated category in question to have additional structure, such as a Noetherian ring action or else a tensor triangulated structure. In this talk we will use pointless topology to develop a theory of supports for any triangulated category whose thick subcategories form a set. To do this, we identify a collection of thick subcategories which are in bijection with the open sets of a topological space.
The study of a space via the lattice of open sets is called pointless topology. Since many topological spaces are completely determined by their lattice of open sets, every topological concept has a pointless, lattice theoretic analogue. Therefore, we can use pointless topology to study the lattice of thick subcategories of a triangulated category from a topological and geometric perspective.

15:30, Room C01142
Markus Schmidmeier (Boca Raton): Finite direct sums of cyclic embeddings with an application to invariant subspace varieties
Abstract: In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. We use partial maps on LittlewoodRichardson tableaux to generalize this result to finite direct sums of such embeddings. As an application to invariant subspaces of nilpotent linear operators, we develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order. This is a report about a joint project with Justyna Kosakowska from Torun.
Friday, 20 May 2016

13:15, Room C01142
Andreas Hochenegger (Köln): Spherical subcategories
Abstract: In a triangulated category, a spherical object is defined as a CalabiYau object that has a twodimensional (graded) endomorphism ring. They are interesting as the associated twist functor gives an autoequivalence. In this talk, I will show what happens if one drops the CalabiYau property, illustrated by examples.
This is joint work with Martin Kalck and David Ploog.

14:30, Room C01142
David Ploog (Berlin): Discrete triangulated categories
Abstract: The study of discretederived algebras (in Vossieck's sense) exhibited some curious properties of their derived categories. E.g. dimensions of homomorphism spaces between indecomposable objects are at most 2; any two objects have only finitely many different cones; hearts of bounded tstructures have only finitely many indecomposable objects. In this talk, we look at such properties among abstract triangulated categories. (Joint work with N. Broomhead and D. Pauksztello.)

16:00, Room C01142
RagnarOlaf Buchweitz (Toronto): Tilting theory for onedimensional Gorenstein algebras
Abstract: We show that for a connected, commutative, positively graded Gorenstein algebra R of Krull dimension one wth nonnegative ainvariant there are tilting objects both for per(qgr R), the triangulated category of perfect complexes of “sheaves” on the (virtual) projective scheme underlying R, as well as for the (larger) stable category of graded maximal CohenMacaulay that are generically locally free.
We’ll discuss in some detail the examples of (not necessarily reduced) line configurations in the plane, the simple curve singularities, and the curve singularities defined by symmetric numerical semigroups.
This is based on joint work with Osamu Iyama and Kota Yamaura.
Friday, 13 May 2016

14:15, Room C01142
Theo Raedschelders (Brussels): Derived categories of noncommutative quadrics and Hilbert schemes of points
Abstract: A philosophy emerging from recent work of Orlov says roughly that for a smooth projective variety X, there should be a smooth projective M_X representing a moduli problem on X such that PerfX embeds as an admissible subcategory into PerfM_X. Moreover, noncommutative deformations of X should embed into commutative deformations of M_X. I will discuss this philosophy and make it precise for X a smooth quadric surface and M_X the Hilbert scheme of two points on X. This is joint work with Pieter Belmans.

15:30, Room C01142
Tobias Barthel (Bonn): Algebraic approximations to stable homotopy theory
Abstract: Viewing the stable homotopy category as a homotopical analogue of the derived category of abelian groups reveals an infinite tower of "chromatic primes" K(n,p) interpolating between characteristic 0 and characteristic p. There are many examples of phenomena in the corresponding K(n,p)local categories that become more algebraic and homogeneous when p goes to infinity. After reviewing the required background from stable homotopy theory, I will explain joint work in progress with Schlank and Stapleton in which we construct an algebraic category that captures such generic phenomena in chromatic homotopy theory. Our methods are inspired by ideas from mathematical logic, and might be applicable in other contexts as well.
Friday, 06 May 2016

14:15, Room C01142
Rolf Farnsteiner (Kiel): Indecomposable Modules, McKay Quivers, and Ramification
Abstract: Let $k$ be an algebraically closed field of characteristic p\ge 3. In 1991, A. Premet determined the Green ring of the restricted enveloping algebra U_0(sl(2)) and provided an explicit description of the indecomposable U_0(sl(2))modules. Earlier work by Drozd, Fischer and Rudakov had essentially shown that the nonsimple blocks of U_0(sl(2)) are Morita equivalent to the trivial extension of the path algebra of the Kronecker quiver. This implies in particular that U_0(sl(2)) is an algebra of domestic representation type. In this talk we indicate how Premet's classification can be extended to finite group schemes of domestic representation type. The combinatorial data of the stable AuslanderReiten quiver of such group schemes are related to McKay quivers and the ramification indices associated to morphisms between certain support varieties.
Friday, 22 April 2016

13:15, Room C01142
Jan Geuenich (Bonn): Jacobian Algebras for Modulated Quivers and Triangulated Orbifolds
Abstract: To begin with, I discuss modulations for weighted quivers in a general framework. After that, I move on to cyclic Galois modulations. I explain what form Jacobian algebras and DWZ mutation assume in this context. As an interesting application I call attention to Jacobian algebras for adjacency quivers of triangulated unpunctured orbifolds. This is joint work with Daniel Labardini Fragoso.

14:30, Room C01142
Oriol RaventosMorera (Regensburg): Generators and descent in triangulated categories
Abstract: The existence of a generator in a triangulated category has strong consequences. Most importantly, it is a fundamental assumption for proving representability results, which in their turn are used to show the existence of adjoint functors and duality formulas.
In this talk, we briefly introduce different notions of generators and exhibit some new examples, especially in the case of derived categories of rings. Next we introduce the notion of decent in a triangulated category and show how it is related to the notion of generator. We explain how descent in triangulated categories can be viewed as an analogue of Grothendieck faithfully flat descent once we work with an infinity categorical enhancement of our triangulated category.

16:00, Room C01142
Peter Symonds (Manchester): Endotrivial modules for infinite groups
Abstract: Endotrivial modules for finite groups have been extensively studied, Here we see what we can say for infinite groups. First, we have to decide on a stable category and work out for which groups it has good properties; Gorenstein projective modules appear extensively here. Then we develop some tools that can be used for calculation in some particular cases.
Friday, 15 April 2016

14:15, Room C01142
Paolo Stellari (Milano): Uniqueness of dg enhancements in geometric contexts
Abstract: It was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper. In this talk we will explain how to extend LuntsOrlov's results to several interesting geometric contexts. Namely, we care about the category of perfect complexes on noetherian separated schemes with enough locally free sheaves and the derived category of quasicoherent sheaves on any scheme. These results will be compared to the existence and uniqueness of dg lifts of exact functors of geometric nature. This is a joint work with A. Canonaco.
Friday, 08 April 2016

14:15, Room V3201
Chrysostomos Psaroudakis (Trondheim): Realisation Functors in Tilting Theory
Abstract: Let T be a triangulated category and H the heart of a tstructure in T. In this setting it is natural to ask what is the relation of T with the bounded derived category of the abelian category H. Under some assumptions on T and the tstructure, BeilinsonBernsteinDeligne constructed a functor between these two triangulated categories, called the realisation functor. The first part of this talk is devoted to recall this construction. Then the main aim is to show how to obtain derived equivalences between abelian categories from not necessarily compact tilting and cotilting objects. The key ingredients of this result are the realisation functor and a notion of (co)tilting objects in triangulated categories that we introduce. As a particular case we explain how derived equivalences between Grothendieck categories can be realised as cotilting equivalences. This is joint work with Jorge Vitoria (arXiv:1511.02677).
For information on earlier talks please check the complete seminar archive.