Seminar

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Seminar Archive

Friday, 12 July 2024

• 13:15, Room V2-200
  Eduardo Vital (Bielefeld): Matroids and Euler characteristics of quiver Grassmannians
  Abstract: We introduce morphisms of matroids with coefficients, which leads to a categorical framework for Baker-Bowler theory. Inspired by the idea that matroids are linear subspaces of F1-vector spaces, we construct quiver Grassmannians of matroids for quiver representations over F1. It turns out that in "nice" cases, the cardinality of F1-rational points (in a suitable sense) of a matroid quiver Grassmannian and the Euler characteristic of its associated complex variety are the same. This is a joint work with Manoel Jarra and Oliver Lorscheid.
  

Friday, 05 July 2024

• 13:15, Room V2-200
  Shiquan Ruan (Xiamen): A geometric model for weighted projective lines of domestic type
  Abstract: In this talk we will give a geometric model for the category of coherent sheaves over the weighted projective lines of domestic type (p,q) and (2,2,n) in terms of an infinite marked strip. We establish a bijection between indecomposable sheaves over the weighted projective lines and certain homotopy classes of oriented curves, and prove that the dimension of extension groups between indecomposable sheaves equals the positive intersection number between the corresponding curves.
  As applications, we provide combinatorial interpretations for the titling graph of tilting bundles, the Picard group action, vector bundle duality, projective cover and injective hull of extension bundles, etc. This is joint work with Jianmin Chen and Hongxia Zhang and Jinfeng Zhang.

Friday, 28 June 2024

• 13:15, Room V2-200
  Edith Hübner (Münster): Animated lambda-rings and Frobenius lifts
  Abstract: We recall Grothendieck’s notion of a lambda-ring and then introduce an infinity-categorical extension of ordinary lambda-rings: animated lambda-rings. As our main result, we prove a characterization of animated lambda-rings in terms of animated rings equipped with a family of coherently compatible Frobenius lifts. In particular, this provides a new perspective on classical lambda-rings which inherently depends on the notion of higher homotopy. We build on Bhatt and Lurie’s results on animated delta-rings in the context of absolute prismatic cohomology and review the necessary infinity-categorical prerequisites during the talk.
• 14:30, Room V2-200
  Severin Barmeier (Köln): Derived equivalences of associative algebras via Fukaya categories
  Abstract: By work of Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk there is a correspondence between derived categories of gentle algebras and partially wrapped Fukaya categories of surfaces with boundary. This correspondence allows one to study derived categories of gentle algebras by their geometric surface models. For example, one may classify indecomposable objects of the derived category in terms of curves on the surface. I will present a generalization of this correspondence to surfaces with orbifold singularities. Their Fukaya categories admit a description by A-infinity algebras whose higher structures can be given explicitly. Using the orbifold surface it is possible to characterize when the Fukaya category is equivalent to the derived category of an associative algebra. This gives a geometric framework for studying derived equivalences between certain classes of associative algebras including skew-gentle algebras and algebras of type D. This is based on joint work with Sibylle Schroll and Zhengfang Wang.
  

Friday, 21 June 2024

• 13:15, Room V2-200
  Victor Torres-Castillo (Ankara): Quantum nonlocal games and the d-torsion commutative space
  Abstract: Nonlocal games have played a prominent role in quantum information theory by demonstrating the power of entanglement. In particular, the 'magic' examples due to Mermin and Peres belong to the class of linear system games. The Mermin-Peres games have no classical solutions, but they admit operator solutions.
  In this talk, we translate the problem of finding operator solutions into a problem of extensions for partial groups (in the sense of Broto-Gonzalez). In particular, we define the d-torsion commutative nerve for groups, whose homotopy structure is crucial to identify a practical criterion (in terms of higher limits) to test a conjecture due to Chung-Okay-Sikora regarding linear system games over Z_d, with d odd.
  This is joint work in progress with Ho Yiu Chung and Cihan Okay.

Friday, 31 May 2024

• 13:15, Room V2-200
  Kevin Schlegel (Stuttgart): Exact structures and purity
  Abstract: We relate the theory of purity of a locally finitely presented category with products to the study of exact structures on the full subcategory of finitely presented objects. Moreover, we focus on the case of a module category over an Artin algebra.

Friday, 24 May 2024

• 13:15, Room V2-200
  Stefan Dawydiak (Bonn): Central extensions in Lusztig's asymptotic Hecke algebra, lower modifications, and tempered representations
  Abstract: Lusztig's asymptotic Hecke algebra J is a based ring whose structure reflects the one- and two-sided cells of an affine (or finite) Weyl group. In the late 80s, Lusztig conjectured that it could be realized as the equivariant K-theory of the square of a finite set, and Bezrukavnikov and Ostrik proved a weak version of this statement in the early 2000s. In joint work with Bezrukavnikov and Dobrovolska, we gave an unexpected counterexample showing that the weak version is in fact optimal. We will explain this, and also give a conceptual explanation for this counterexample via the relationship between J and representations of p-adic groups given by Braverman and Kazhdan.

Friday, 03 May 2024

• 13:15, Room V2-200
  Anish Chedalavada (Baltimore): A derived refinement of a classical theorem in tt-geometry
  Abstract: In this talk, we explain how to equip the Balmer spectrum of a rigidly-compactly generated symmetric monoidal stable infinity-category with a natural structure sheaf, generalizing gluing techniques of Balmer-Favi in a systematic fashion. Following tensor-triangular philosophy in porting over statements from algebraic geometry, we provide a universal property for this equipment (in the style of an affine scheme). As an application, we explain how to recover the computation of the thick tensor ideals of the perfect complexes on a qcqs scheme, in addition to partially generalizing the statement to nice classes of spectral stacks. In another direction, we explain how one might utilise these techniques to import methods from spectral algebraic geometry into a wider variety of contexts.
  

Friday, 26 April 2024

• 13:15, Room V2-200
  Sven-Ake Wegner (Hamburg): The two derived categories of the LB-spaces
  Abstract: Let LB be the category of LB-spaces, which has as objects precisely those Hausdorff locally convex spaces that can be written as a countable inductive limit of Banach spaces, and as morphisms the continuous linear maps between them. In the talk we will firstly review LBs categorical properties and explain its place in the general hierarchy of non-abelian categories. After that we will show that there are (at least) two natural, but not naturally equivalent, ways to define a derived category of LB.
  

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.
  
  

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