No talks have been announced for this week.

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Future Talks

Friday, 12 April 2024

  • 13:15
    Sven-Ake Wegner (Hamburg): tba

Seminar Archive

Friday, 02 February 2024

  • 13:15, Room T2-233
    Panagiotis Kostas (Thessaloniki): Injective generation for tensor rings
    Abstract: In 2019 Rickard introduced a condition for rings, called injective generation and proved that if injectives generate for a finite dimensional algebra, then this algebra has finite finitistic dimension. After discussing some graded aspects of injective generation, we will prove that given a ring R and a "sufficiently nice" R-bimodule M, then injectives generate for R if and only if injectives generate for the tensor ring of R by M. This is based on joint work with Chrysostomos Psaroudakis.

Friday, 15 December 2023

  • 13:15, Room T2-233
    Lara Bossinger (Oaxaca): Brick compactifications of braid varieties using superpotentials
    Abstract: Braid varieties have gained interest recently with the recovery of their cluster structures. A natural compactification of braid varieties to brick varieties was given a few years ago by Escobar. I will explain how the brick compactification of braid varieties can be obtained in the context of Gross-Hacking-Keel-Kontsevich superpotentials for cluster varieties. This is joint work in progress based on an AIMS working group formed by José Simental, Daping Weng, Iva Halacheva, Allen Knutson, Pavel Galashin et al.
  • 14:30, Room T2-233
    David Ploog (Stavanger): The heart fan of a triangulated category
    Abstract: I will discuss a general construction attaching (a) a convex cone to an abelian category, (b) a fan to a bounded heart in a triangulated category and (c) a multifan to a triangulated category. These constructions generalise the g-fan of a finite-dimensional algebra. If the bounded heart is length then our fan is always complete; in particular, it provides a natural completion of the g-fan. These constructions are motivated by, and lead to a convex-geometric description of, the stability space of a triangulated category.
    (Joint work with Nathan Broomhead, David Pauksztello, Jon Woolf.)

Friday, 24 November 2023

  • 13:15, Room T2-233
    Chris Parker (Bielefeld): Bounded t-structures, finitistic dimensions, and singularity categories of triangulated categories
    Abstract: We will talk about recent joint work on a triangulated categorical generalisation of Neeman's theorem on the existence of bounded t-structures on the derived category of perfect complexes, which solved a bold conjecture by Antieau, Gepner, and Heller. In particular, under mild conditions, we show how the existence of a bounded t-structure on a triangulated category implies that its singularity category vanishes. To achieve this, we show that certain t-structures can be lifted from a triangulated category to its completion, as well as introduce the notion of finitistic dimension for triangulated categories. This work is joint with Rudradip Biswas, Kabeer Manali Rahul, Hongxing Chen, and Junhua Zheng.
  • 14:30, Room T2-233
    Kyungmin Rho (Paderborn): Homological mirror symmetry correspondence on an affine model
    Abstract: The affine normal crossing surface singularity xyz=0 (B-model) is a mirror dual of the three-punctured Riemann sphere (A-model). On the B-model, we consider the stable category of maximal Cohen-Macaulay (a.k.a. Gorenstein projective) modules, whose indecomposable isomorphism classes have been completely classified by Burban-Drozd's representation-theoretic method. We find their corresponding curves in the Fukaya category of the A-model and show their one-to-one correspondence with immersed geodesics. We also explain some interchanges of algebraic operations and geometric symmetries and discuss how to globalize this local aspect for more general mirror pairs. This is based on joint works with Cheol-Hyun Cho, Wonbo Jeong, and Kyoungmo Kim.

Friday, 10 November 2023

  • 13:15, Room T2-233
    Umesh V. Dubey (Prayagraj): Tensor t-structures on the derived category of a Noetherian scheme
    Abstract: The notion of truncation structure (t-str) was introduced in the famous paper of Beilinson, Bernstein, and Deligne (Gabber). It has found applications in various areas.
    We will discuss the notion of tensor-compatible t-str (or tensor t-str) on the derived category of schemes (w.r.t. some fixed t-str). The notion of tensor t-structure is motivated by the classification of thick tensor ideal subcategories of derived categories of schemes. Thomason used tensor to extend the known classification theorem of Hopkins-Neeman from affine case to more general schemes.
    We will describe our classification of compactly generated tensor t-structures on the derived category of Noetherian schemes in terms of Thomason filtrations. It extends the known classification results on the derived category of Noetherian ring. As an application, we can prove the tensor telescope conjecture for t-structures in the sense of Hrebek.
    This is based on the joint work with Gopinath Sahoo.

Friday, 27 October 2023

  • 13:15, Room T2-233
    Calvin Pfeifer (Odense): On τ-tilting tameness of affine GLS algebras
    Abstract: Geiß-Leclerc-Schröer (GLS) associated to every valued quiver Γ a finite-dimensional algebra H defined in terms of quivers with relations. Their algebras H are 1-Iwanaga-Gorenstein and arise as degenerations of hereditary algebras. These degenerations are representation tame whenever the valued quiver Γ is affine. In contrast, corresponding GLS algebras are often representation wild. This raises the question in which sense affine GLS are still „tame“. In this talk, we present a generic classification of locally-free representations of affine GLS algebras. We deduce that affine GLS algebras are „tame“ from the perspective of τ-tilting theory.
    An integral part of our generic classification is the construction of a 1-parameter family of representations stable with respect to the defect. In particular, we verify τ-tilting versions of the second Brauer-Thrall conjecture introduced by Mousavand, for the class of GLS algebras.
  • 14:30, Room T2-233
    Kaveh Mousavand (Okinawa): Distribution of bricks -- algebraic and geometric viewpoints
    Abstract: In a series of joint work with Charles Paquette, we have studied the behaviour of bricks from different perspectives. More specifically, for a (basic) finite dimensional associative algebra A over an algebraically closed field, we are mainly concerned with the behaviour of those A-modules whose endomorphism algebras are division rings. Every such module is called a brick. Our work is primarily motivated by a conjecture that I posed in 2019, which concerns the distribution of finite dimensional bricks: An algebra A admits infinitely many isomorphism classes of bricks if and only if A admits an infinite family of bricks of the same length. In this talk, I present some of our new results on this (still open) conjecture, as well as the connections to the infinite dimensional bricks and generic modules. Then, I will discuss some interesting applications of our results in the study of stability conditions and tau-tilting theory.

Friday, 13 October 2023

  • 13:15, Room T2-233
    Juan Omar Gomez (Bielefeld): Enhanced stable categories for infinite groups and applications
    Abstract: Informally, the stable module category for an infinite group over a field of positive characteristic is obtained from the category of modules over the group ring by discarding modules of finite projective dimension. In this talk we will introduce an enhancement of the stable module category for infinite groups, and we will present two applications of this approach: first, we provide a formula to classify invertible modules in the stable module category for an infinite group with a finite dimensional cocompact model for the classifying space for proper actions; and second, we construct a family of infinite degree separable commutative algebras giving a negative answer to an open question by P. Balmer.

Saturday, 30 September 2023

Friday, 29 September 2023

Thursday, 28 September 2023

Wednesday, 27 September 2023

Tuesday, 26 September 2023

Monday, 25 September 2023

Friday, 07 July 2023

  • 13:15, Room U2-113
    Grzegorz Bobiński (Torun): Categories of filtered modules
    Abstract: For representation directed algebras deformations of a module N are filtered by indecomposable direct summands of N. We present a construction of a category, being a variation of the category of modules filtered by indecomposable direct summands of N, which controls geometry of deformations of N. We prove basic properties of this category. This is a report on a joint work in progress with Grzegorz Zwara.
  • 14:30, Room U2-113
    Wassilij Gnedin (Paderborn): Gluing two-term silting complexes over certain pullback rings
    Abstract: In 1971, Milnor described a technique to glue projective modules over rings which arise as the pullback of a surjective and an arbitrary ring homomorphism.
    More recently, Burban and Drozd developed a variation of the gluing technique in order to classify the indecomposable objects in the right-bounded derived category of a skew-gentle algebra. Further examples of pullback rings of the form above are given by radical square zero algebras and Bäckström orders.

    A decade ago, Adachi, Iyama and Reiten showed that two-term silting complexes (or, in an equivalent formulation, support tau-tilting modules) provide a natural extension of classical tilting theory. Since then, the problem to classify two-term silting complexes over a given algebra has attracted a lot of interest.

    The goal of my talk is to provide an adaptation of the gluing technique, which allows to reduce the classification problem of two-term silting complexes over a Bäckström algebra to the study of rigid modules over a certain finite-dimensional algebra.

Friday, 30 June 2023

  • 13:15, Room U2-113
    Aran Tattar (Köln): Weak stability conditions and the space of chains of torsion classes
    Abstract: Joyce introduced the concept of weak stability conditions for an abelian category as a generalisation of Rudakov’s stability conditions. We show an explicit relationship between chains of torsion classes and weak stability conditions. In particular, up to a natural equivalence, they coincide. We also discuss topological properties of the space of chains of torsion classes and its quotient given by this equivalence relation.

Wednesday, 21 June 2023

  • Bielefeld-Paderborn Representation Theory Seminar
    14:30, Room J3.330 (Paderborn)
    Okke van Garderen (Luxembourg): The hidden symmetries of McKay quivers
    Abstract: A famous correspondence of McKay relates the resolution of the quotient singularity associated to a finite subgroup of SU(2) with a quiver algebra in the shape of a Dynkin diagram.
    Looking at the moduli spaces of this quiver algebra, one stumbles upon a natural relation with root systems and finds various symmetries coming from the associated Weyl group.
    This talk will be about the McKay quivers coming from subgroups of SU(3), whose moduli spaces determine interesting invariants of Calabi-Yau threefolds. Although these quivers are not shaped like Dynkin diagrams, I will explain how one can nonetheless identify a 'hidden' symmetry coming from a type of Dynkin combinatorics.
  • Bielefeld-Paderborn Representation Theory Seminar
    16:00, Room J3.330 (Paderborn)
    Sondre Kvamme (Trondheim): Indecomposables in the monomorphism category
    Abstract: This is joint work with Nan Gao, Julian Külshammer and Chrysostomos Psaroudakis.
    The study of submodule categories is an important and old subject in representation theory. It has connections to, for example, Littlewood—Richardson tableaux, valuated p-groups and metabelian groups. In 2004 Ringel and Schmidmeier studied such categories using modern tools like Auslander—Reiten theory and covering theory.
    A generalization of submodule categories, called (separated) monomorphism categories, has been more recently studied. It has connections to cotorsion pairs, Gorenstein homological algebra, singularity theory and topological data analysis.
    In this talk I will define submodule categories and monomorphism categories. Then I will explain how they can be related to representations over stable categories via epivalences (also called representation equivalences), and how this can often be used to determine their indecomposables. I will also say something about our proof of this, which uses free monads on abelian categories. If time permits, I will discuss analogues of monomorphism categories for species. In particular, I will explain how our result can be used to give a characterization of Cohen-Macaulay finiteness for the algebras H associated to symmetrizable Cartan matrices introduced by Geiss-Leclerc-Schröer, assuming the terms in the symmetrizer are less than or equal to 2.
  • Bielefeld-Paderborn Representation Theory Seminar
    17:15, Room J3.330 (Paderborn)
    Karin Erdmann (Oxford): Tame symmetric algebras
    Abstract: This will review Hybrid Algebras (introduced in joint work with Andrzej Skowronski), and discuss whether these might be 'almost all' tame symmetric algebras.

Friday, 16 June 2023

Thursday, 15 June 2023

Friday, 09 June 2023

  • 13:15, Room U2-113
    Paolo Stellari (Milano): Deformations of stability conditions with applications to Hilbert schemes of points and very general abelian varieties
    Abstract: The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk I will illustrate a new and very effective technique based on deformations. A key ingredient is a general result about deformations of bounded t-structures (with some additional and mild assumptions). Two remarkable applications are the construction of stability conditions for very general abelian varieties in any dimension and for some irreducible holomorphic symplectic manifolds, again in all possible dimensions. This is joint work with C. Li, E. Macri' and X. Zhao.
  • 14:30, Room U2-113
    Amnon Neeman (Canberra): On generalizations of an old theorem of Rickard's
    Abstract: In 1989 Rickard published a couple of papers on Morita equivalence of derived categories. We will begin with a quick review of his results.
    The interest in this old result was revived in 2018 by Krause, who asked for improvements. Krause's question almost immediately led to a couple of results we will briefly review.
    And then we will talk about very recent work, joint with Canonaco and Stellari, which goes much further in improving Rickard's theorem. This is work still in progress, in the sense that we are trying to sharpen and generalize the theorems that we can already prove.
  • 16:00, Room U2-113
    Wilberd van der Kallen (Utrecht): Reductivity and finite generation
    Abstract: Recall that the first fundamental theorem of invariant theory is about finite generation of the subalgebra of invariants, when a reductive group acts on an algebra. We replace the group with an affine group scheme G and also ask about finite generation of the G-cohomology of the algebra.

Friday, 26 May 2023

  • 13:15, U2-113
    Andrea Solotar (Buenos Aires): Strong stratifying Morita contexts
    Abstract: We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context (after H. Bass) is an algebra built on the data of two algebras, two bimodules and two morphisms. For a strong stratifying Morita context - or equivalently for a strong stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. This is a work in collaboration with Claude Cibils, Marcelo Lanzilotta and Eduardo Marcos.

Wednesday, 17 May 2023

Friday, 12 May 2023

  • 13:15, Room U2-113
    Markus Schmidmeier (Boca Raton): Invariant subspaces of nilpotent linear operators: Level, mean and colevel
    Abstract: Let S(n) be the category of all pairs X=(U,V) where V is a finite dimensional vector space with a nilpotent linear operator T which satisfies T^n=0, and where U is a T-invariant subspace of V. Note that S(n) is just the category of Gorenstein-projective modules over the triangular matrix ring with coefficients in k[T]/T^n.
    We consider three invariants for a non-zero pair X: If u, v and w denote the dimensions of U, V and V/U and b is the number of Jordan-blocks of V, then the level, mean and colevel of X are p=u/b, q=v/b and r=w/b.
    The pr-vector of X is just the pair (p,r); together, the pr-vectors of the indecomposable objects in S(n) form the pr-picture.
    In my talk I discuss properties of level, mean and colevel, and how the pr-picture evolves as n increases.
    This is a report about recent work with Claus Michael Ringel.

Friday, 05 May 2023

  • 13:15, Room U2-113
    Johannes Krah (Bielefeld): Phantoms and exceptional collections on rational surfaces
    Abstract: A smooth projective rational surface over an algebraically closed field admits a full exceptional collection. Building on work of Hille--Perling, Perling, and Vial, we study mutations of (numerically) exceptional collections by analyzing the lattice theoretic behavior of the numerical Grothendieck group. On the one hand, we show that some results, known for del Pezzo surfaces, can be extended to the blow-up of the projective plane in 9 points in very general position. On the other hand, on the blow-up of 10 points in general position we construct an exceptional collection of maximal length which is not full. This disproves a conjecture of Kuznetsov and a conjecture of Orlov. As an application in representation theory, our example shows that the derived equivalent finite dimensional algebra carries a presilting object which is not a direct summand of a silting object.
  • 14:30, Room U2-113
    Lutz Hille (Münster): Polynomial invariants for triangulated categories with full exceptional sequences
    Abstract: For a full exceptional sequence of vector bundles on the projective plane there is a remarkable equation, the so-called Markov equation, in terms of the ranks of the three vector bundles. This equation, slightly modified, has been used in a joint work with Beineke and Brüstle for cluster mutations for quivers with three vertices.
    The aim of this talk is to define the natural generalization for full exceptional sequences with n members. This leads to the notion of a polynomial invariant, that is a polynomial in indeterminants x(i,j) for i < j between 1 and n. This allows to evaluate such a polynomial at any full exceptional sequence. We define a polynomial invariant to be a polynomial whose value does not depend on the full exceptional sequence, it only depends on the underlying category.
    In the talk we define polynomial invariants, present several examples and relate them to the natural braid group action. Eventually, we classify all polynomial invariants.

Wednesday, 03 May 2023

Friday, 28 April 2023

  • 13:15, Room U2-113
    Chiara Sava (Prague): ∞-Dold-Kan correspondence via representation theory
    Abstract: Both Happel and Ladkani proved that, for commutative rings, the quiver An is derived equivalent to the diagram generated by An where any composition of two consecutive arrows vanishes. We give a purely derivator-theoretic reformulation and proof of this result showing that it occurs uniformly across stable derivators and it is then independent of coefficients. The resulting equivalence provides a bridge between homotopy theory and representation theory; in fact we will see how our result is a derivator-theoretic version of the ∞-Dold-Kan correspondence for bounded cochain complexes.

Wednesday, 26 April 2023

Friday, 21 April 2023

  • 13:15, Room U2-113
    Edmund Heng (Bures-sur-Yvette): Quiver representations in fusion categories; extending Gabriel’s theorem to ABCDEFG and HI!
    Abstract: One of the most celebrated theorem in quiver representations is undoubtedly Gabriel’s theorem, which reveals a deep connection between quiver representations and root systems arising from Lie algebras. It shows that the finite-type quivers are classified by the ADE Dynkin diagrams and the indecomposable representations are in bijection with the underlying positive roots. Following the works of Dlab—Ringel, the classification can be generalised to include all the other Dynkin diagrams (BCFG) by considering the more general notion of valued quivers (K-species) representations instead.
    However, from the point of view of Coxeter theory, which also have well-defined root systems extending those of ADE types, the (non-crystallographic) types H and I are missing from the classification. The aim of this talk is to introduce a new notion of representations for a class of (labelled) quivers known as Coxeter quivers, where their representations are built in certain fusion categories. The relevance to Gabriel’s theorem is then as follows: a Coxeter quiver has finitely many indecomposable representations if and only if its underlying graph is a Coxeter-Dynkin diagram — including types H and I. Moreover, the indecomposable representations of a Coxeter quiver are in bijection with extended the positive roots of the underlying root system of the Coxeter group.

Wednesday, 19 April 2023

Friday, 14 April 2023

  • 13:15, Room U2-113
    Alexander Pütz (Bochum): GKM Theory for Quiver Grassmannians
    Abstract: Projective varieties have various realisations as quiver Grassmannians. Finding good realisations and nice torus actions on the corresponding quiver Grassmannians allows to compute torus equivariant cohomology using GKM theory. So far we have established nice torus actions on quiver Grassmannians for equioriented quivers of type A and nilpotent representations of the equioriented cycle. This leads to a combinatorial description of torus fixed points and one-dimensional orbits. We believe that a similar construction also works for other quivers.

For information on earlier talks please check the complete seminar archive.