Seminar
No talks have been announced for this week.
For a regular email announcement please contact birep.
Future Talks
Friday, 21 October 2016
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Michael K. Brown (Bonn): Topological K-theory of dg categories of graded matrix factorizations
Abstract: Topological K-theory of complex-linear dg categories is a notion recently introduced by A. Blanc. The main goal of the talk is to discuss a calculation of the topological K-theory of the dg category of graded matrix factorizations associated to a complex quasi-homogeneous 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, 28 October 2016
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Sven-Ake Wegner (Wuppertal): tba
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Gustavo Jasso (Bonn): tba
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Julian Külshammer (Stuttgart): tba
Seminar Archive
Friday, 26 August 2016
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14:15, Room V5-227
Zhi-Wei Li (Xuzhou): A homotopy theory of additive category with suspensions
Abstract: We give a definition of partial one-sided triangulated categories. We show that complete cotorsion pairs in exact categories, torsion pairs and mutation pairs in triangulated categories all extend to partial one-sided triangulated categories. We prove that partial one-sided triangulated categories yield one-sided 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 Koenig-Zhu, Keller-Reiten and Kussin-Lenzing-Meltzer. The second one is the construction of stable triangulated categories which allows us to model Iyama-Yoshino subfactors of triangulated categories via Quillen closed model structures. The last one is to develop a homotopy theory of additive categories with suspensions via Gabriel-Zisman localization which leads to a Buchweitz type theorem in triangulated categories. This theorem extends the recent work of Wei and Iyama-Yang which are generalizations of Buchweitz's work on singularity categories. As a corollary we give a triangle equivalence between Verdier quotients and Iyama-Yoshino subfactors of triangulated categories under suitable conditions.
Thursday, 21 July 2016
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16:15, Room V3-204
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^2-pqr, 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
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13:15, Room C01-142
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 Wedderburn-Artin 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 so-called 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.
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14:30, Room C01-142
Martin Kalck (Edinburgh): Knörrer-type equivalences for two-dimensional cyclic quotient singularities
Abstract: We construct triangle equivalences between singularity categories of two-dimensional 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 quasi-hereditary algebras. This is based on joint work with Joe Karmazyn.
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16:00, Room C01-142
Grzegorz Bobiński (Torun): On nonsingularity in codimension one of irreducible components of module varieties over quasi-tilted 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 quasi-tilted algebras.
Friday, 01 July 2016
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13:15, Room C01-142
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 Ringel-Schmidmeier, 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.
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14:30, Room C01-142
Alexander Kleshchev (Eugene): Stratifications of Khovanov-Lauda-Rouquier algebras
Abstract: We review standard module theory for Khovanov-Lauda-Rouquier 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
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14:15, Room C01-142
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
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13:15, Room C01-142
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.
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14:30, Room C01-142
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.
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16:00, Room C01-142
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
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14:15, Room C01-142
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
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14:15, Room C01-142
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.
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15:30, Room C01-142
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 Littlewood-Richardson 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
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13:15, Room C01-142
Andreas Hochenegger (Köln): Spherical subcategories
Abstract: In a triangulated category, a spherical object is defined as a Calabi-Yau object that has a two-dimensional (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 Calabi-Yau property, illustrated by examples.
This is joint work with Martin Kalck and David Ploog.
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14:30, Room C01-142
David Ploog (Berlin): Discrete triangulated categories
Abstract: The study of discrete-derived 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 t-structures 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.)
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16:00, Room C01-142
Ragnar-Olaf Buchweitz (Toronto): Tilting theory for one-dimensional Gorenstein algebras
Abstract: We show that for a connected, commutative, positively graded Gorenstein algebra R of Krull dimension one wth nonnegative a-invariant 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 Cohen-Macaulay 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
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14:15, Room C01-142
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 Perf-X embeds as an admissible subcategory into Perf-M_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.
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15:30, Room C01-142
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
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14:15, Room C01-142
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 non-simple 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 Auslander-Reiten 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
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13:15, Room C01-142
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.
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14:30, Room C01-142
Oriol Raventos-Morera (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.
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16:00, Room C01-142
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
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14:15, Room C01-142
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 Lunts-Orlov'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 quasi-coherent 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
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14:15, Room V3-201
Chrysostomos Psaroudakis (Trondheim): Realisation Functors in Tilting Theory
Abstract: Let T be a triangulated category and H the heart of a t-structure 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 t-structure, Beilinson-Bernstein-Deligne 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
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10:15, Room V2-213
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 Labardini-Fragoso. 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
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14:15, Room V2-213
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
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Faculty Colloquium
17:15, Room V2-210/216
Paul Balmer (Los Angeles): An invitation to tensor-triangular geometry
Abstract: We will begin by an overview of the various fields where tensor-triangulated 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
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13:15, Room V2-213
Rebecca Reischuk (Bielefeld): The adjoints of the Schur functor
Abstract: The so-called 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 well-known 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.
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14:30, Room V2-213
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).
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16:00, Room V2-213
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
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14:15, Room V2-213
Matthew Pressland (Bath): Internally Calabi-Yau algebras and cluster-tilting objects
Abstract: Cluster categories, which are 2-Calabi–Yau triangulated categories containing cluster-tilting 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 2-Calabi-Yau, although such a categorification is only known in a few cases. It is observed by Keller-Reiten that the endomorphism algebra of a cluster-tilting object in such a category has a certain relative, or internal, Calabi-Yau symmetry. In this talk, I will explain how to go in the opposite direction; given an algebra A with a suitable level of Calabi-Yau symmetry, I will explain how to construct a Frobenius category admitting a cluster-tilting object with endomorphism algebra A.
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15:30, Room V2-213
Igor Burban (Cologne): Singular curves and quasi-hereditary algebras
Abstract: In my talk (based on a joint work with Yu. Drozd and V. Gavran), I shall describe a certain non-commutative 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(A-mod) as a full subcategory.
Friday, 15 January 2016
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14:15, Room V2-213
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 quasi-coherent 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
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14:15, Room V2-213
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 non-prime cluster variables.
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15:30, Room V2-213
Henning Krause (Bielefeld): The variety of subadditive functions for finite group schemes
Abstract: For a finite group scheme G, Friedlander and Pevtsova introduced pi-points which give rise to certain endofinite 'point modules'. Using then Crawley-Boevey'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
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14:15, Room V2-213
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.
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15:30, Room V2-213
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 two-sided 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
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14:15, Room V2-213
Hans Franzen (Bonn): Donaldson-Thomas 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
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13:15, Room V2-213
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 skew-symmetric cluster algebras with coefficients. As a consequence we obtain a description of the category of Cohen-Macaulay modules over certain isolated singularities as the completed orbit category of a Nakajima category.
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14:30, Room V2-213
Magnus Engenhorst (Bonn): Maximal green sequences for quiver categories
Abstract: Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic 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 t-structures 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.
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16:00, Room V2-213
Raymundo Bautista (Morelia): Differential tensor algebras, boxes, and exact structures
Abstract: A box B consists of an algebra A over some field, an A-A bimodule U with a co-associative co-multiplication, and a co-unit. 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 tame-wild dichotomy proved by Y. Drozd and the discovery due to W. Crawley-Boevey 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 A-A bimodule given by the kernel of the co-unit (an A-A 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 quasi-hereditary algebras and the category of modules with standard filtration.
Friday, 13 November 2015
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14:15, Room V2-213
Thomas Gobet (Kaiserslautern): On twisted filtrations on Soergel bimodules
Abstract: The Iwahori-Hecke 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 Kazhdan-Lusztig positivity.
Friday, 06 November 2015
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14:15, Room V2-213
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 conjugation-action 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, Delta-filtrations and more. (This is work in progress, joint with M. Bulois)
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15:30, Room V2-213
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 non-trivial p-subgroups. Some time will be spent introducing the Brown complex and related basic questions.
Friday, 30 October 2015
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14:15, Room V2-213
Dirk Kussin (Paderborn): Infinite-dimensional 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 pure-injective modules.
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15:30, Room V2-213
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 Hom-finite 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 ad-bc=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 quasi-coherent sheaves (resp. indecomposable infinite dimensional B-modules) of irrational slope r, a problem attacked by Harland and Prest during the last years through model-theoretic methods.
Friday, 23 October 2015
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14:15, Room V2-213
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 Ringel-Schmidmeier 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.
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15:30, Room V2-213
Julia Sauter (Bielefeld): On quiver Grassmannians and orbit closures for representation-finite algebras
Abstract: We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra.
For any representation-finite 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 Crawley-Boevey and it generalizes results of Cerulli Irelli, Feigin and Reineke.
For information on earlier talks please check the complete seminar archive.
