Seminar
Friday, 21 June 2024

13:15, Room V2200
Victor TorresCastillo (Ankara): Quantum nonlocal games and the dtorsion 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 MerminPeres 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 BrotoGonzalez). In particular, we define the dtorsion 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 ChungOkaySikora regarding linear system games over Z_d, with d odd.
This is joint work in progress with Ho Yiu Chung and Cihan Okay.
For a regular email announcement please contact birep.
Future Talks
Friday, 28 June 2024

13:15, Room V2200
Edith Hübner (Münster): Animated lambdarings and Frobenius lifts
Abstract: We recall Grothendieck’s notion of a lambdaring and then introduce an infinitycategorical extension of ordinary lambdarings: animated lambdarings. As our main result, we prove a characterization of animated lambdarings in terms of animated rings equipped with a family of coherently compatible Frobenius lifts. In particular, this provides a new perspective on classical lambdarings which inherently depends on the notion of higher homotopy. We build on Bhatt and Lurie’s results on animated deltarings in the context of absolute prismatic cohomology and review the necessary infinitycategorical prerequisites during the talk. 
14:30, Room V2200
Severin Barmeier (Köln): Derived equivalences of associative algebras via Fukaya categories
Abstract: By work of HaidenKatzarkovKontsevich and LekiliPolishchuk 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 Ainfinity 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 skewgentle algebras and algebras of type D. This is based on joint work with Sibylle Schroll and Zhengfang Wang.
Friday, 12 July 2024

13:15, Room V2200
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 BakerBowler theory. Inspired by the idea that matroids are linear subspaces of F1vector spaces, we construct quiver Grassmannians of matroids for quiver representations over F1. It turns out that in "nice" cases, the cardinality of F1rational 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.
Seminar Archive
Friday, 31 May 2024

13:15, Room V2200
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 V2200
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 twosided cells of an affine (or finite) Weyl group. In the late 80s, Lusztig conjectured that it could be realized as the equivariant Ktheory 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 padic groups given by Braverman and Kazhdan.
Friday, 03 May 2024

13:15, Room V2200
Anish Chedalavada (Baltimore): A derived refinement of a classical theorem in ttgeometry
Abstract: In this talk, we explain how to equip the Balmer spectrum of a rigidlycompactly generated symmetric monoidal stable infinitycategory with a natural structure sheaf, generalizing gluing techniques of BalmerFavi in a systematic fashion. Following tensortriangular 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 V2200
SvenAke Wegner (Hamburg): The two derived categories of the LBspaces
Abstract: Let LB be the category of LBspaces, 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 nonabelian 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 T2233
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" Rbimodule 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 T2233
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 GrossHackingKeelKontsevich 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 T2233
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 gfan of a finitedimensional algebra. If the bounded heart is length then our fan is always complete; in particular, it provides a natural completion of the gfan. These constructions are motivated by, and lead to a convexgeometric 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 T2233
Chris Parker (Bielefeld): Bounded tstructures, 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 tstructures 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 tstructure on a triangulated category implies that its singularity category vanishes. To achieve this, we show that certain tstructures 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 T2233
Kyungmin Rho (Paderborn): Homological mirror symmetry correspondence on an affine model
Abstract: The affine normal crossing surface singularity xyz=0 (Bmodel) is a mirror dual of the threepunctured Riemann sphere (Amodel). On the Bmodel, we consider the stable category of maximal CohenMacaulay (a.k.a. Gorenstein projective) modules, whose indecomposable isomorphism classes have been completely classified by BurbanDrozd's representationtheoretic method. We find their corresponding curves in the Fukaya category of the Amodel and show their onetoone 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 CheolHyun Cho, Wonbo Jeong, and Kyoungmo Kim.
Friday, 10 November 2023

13:15, Room T2233
Umesh V. Dubey (Prayagraj): Tensor tstructures on the derived category of a Noetherian scheme
Abstract: The notion of truncation structure (tstr) 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 tensorcompatible tstr (or tensor tstr) on the derived category of schemes (w.r.t. some fixed tstr). The notion of tensor tstructure 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 HopkinsNeeman from affine case to more general schemes.
We will describe our classification of compactly generated tensor tstructures 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 tstructures in the sense of Hrebek.
This is based on the joint work with Gopinath Sahoo.
Friday, 27 October 2023

13:15, Room T2233
Calvin Pfeifer (Odense): On τtilting tameness of affine GLS algebras
Abstract: GeißLeclercSchröer (GLS) associated to every valued quiver Γ a finitedimensional algebra H defined in terms of quivers with relations. Their algebras H are 1IwanagaGorenstein 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 locallyfree representations of affine GLS algebras. We deduce that affine GLS algebras are „tame“ from the perspective of τtilting theory.
An integral part of our generic classification is the construction of a 1parameter family of representations stable with respect to the defect. In particular, we verify τtilting versions of the second BrauerThrall conjecture introduced by Mousavand, for the class of GLS algebras. 
14:30, Room T2233
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 Amodules 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 tautilting theory.
Friday, 13 October 2023

13:15, Room T2233
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.