Program

Zoom link for the conference:
https://universityofleeds.zoom.us/j/87663974774?pwd=ahTZIfNwa41296XqHwiRqyoXyaMnrD.1

Meeting ID: 876 6397 4774
Passcode: @Flash6

All times are in CET (UTC+1).

This talk is based on the joint work 2404.13232 with Osamu Iyama. For an abelian category \(\mathcal{A}\) with only finitely many isoclasses \(S(1),\ldots,S(n)\) of simple objects, its real Grothendieck group \(K_0(\mathcal{A})_\mathbb{R}\) and the dual \(K_0(\mathcal{A})_\mathbb{R}^*\) are both identified with \(\mathbb{R}^n\). We focus on the latter one. In my previous work, I defined the TF equivalence as an equivalence relation on \(K_0(\mathcal{A})_\mathbb{R}^*\) via stability conditions and torsion pairs of King and Baumann-Kamnitzer-Tingley. In this joint work, we introduce the \(M\)-TF equivalence for each object \(M \in \mathcal{A}\) as a systematic way to coarsen the TF equivalence preserving representation-theoretical meaning. I would like to explain the definition and some basic properties of \(M\)-TF equivalences in this talk.

Sota Asai's slides

In type A, for some families of representations of quantum affine algebras, decomposition formulas can be written as matrix determinant. Then, one can get new formulas using the Lewis Carroll identity on these matrix. In specific cases, this recovers the well-known T-systems formulas, or its generalizations, and they can have a cluster algebraic interpretation.

Léa Bittmann's slides

Let \(L:k\) be a field extension, let \(\Lambda\) be a finite-dimensional \(k\)-algebra and let \(\Lambda^L\) denote the finite-dimensional \(L\)-algebra \(\Lambda \otimes_k L\). Jensen and Lenzing (1982) showed that if \(L:k\) is MacLane separable, then \(\Lambda^L\) is representation-finite if and only if \(\Lambda\) is. In informal terms, extension of scalars along "nice" field extensions preserves representation-finiteness. We report on joint work in progress with M. Kaipel, C. Klapproth and K. Schlegel, where we study the more general concept of brick-finiteness, i.e. admitting only a finite number of isomorphism classes of modules whose endomorphism algebra is a division algebra. Our findings are counter-intuitive. For many "nice" choices of \(L:k\), brick-finiteness is not preserved in general. More precisely, there are counter-examples whenever L contains a non-trivial algebraic and separable element over k. Time permitting, we discuss how brick-finiteness can be preserved in certain less "nice" cases.

Erlend D. Børve's slides (Updated)

We show two results related to the three constructions of cluster categories: as orbit categories, as singularity categories and as cosingularity categories. We show the universal property of pretriangulated orbit categories of dg categories first stated by Keller in 2005 and apply this to study collapsing of grading for (higher) cluster categories constructed from bigraded Calabi-Yau completions as introduced by Ikeda-Qiu. We show that for any dg algebra \(A\), its perfect derived category can be realized in two ways: as an (enlarged) cluster category or as a (shrunk) singularity category, generalizing results of Ikeda-Qiu and Happel following Hanihara respectively, and relate these two descriptions using a version of relative Koszul duality. This is a joint work with Bernhard Keller and Yu Qiu.

Li Fan's slides (Updated)

Auslander and Reiten discovered that every Auslander-Gorenstein algebra admits a distinguished bijection between its indecomposable projective and injective modules, now known as the Auslander-Reiten bijection. In this talk, we present a new result showing that, for certain classes of algebras, the existence of such a bijection actually characterises the Auslander-Gorenstein property. We then discuss a new, linear algebraic interpretation of the Auslander-Gorenstein property and the Auslander-Reiten bijection using Coxeter matrices and their Bruhat decompositions, based on a joint work with René Marczinzik and Hugh Thomas. This approach opens the door to extending the definition of the Auslander-Reiten bijection to algebras that are not Auslander-Gorenstein.

Viktória Klász' slides

We investigate the relationship between smoothness and the relative global dimension of a ring extension. We prove that a smooth commutative algebra A over B has finite relative global dimension gldimp(A, B). Conversely, under a mild condition on B, the finiteness of gldimp(A, B) implies that the map B → A is smooth. We also relate the relative global dimension to the usual global dimension of the fibers of B → A, and establish a formula for the relative global dimension of tensor products of extensions. Finally, we present examples and an alternative characterization of smoothness in terms of relative Hochschild homology.

I will explain how to generalise Hochschild cohomology to algebra 1-morphisms in (quasi-multi)fiat 2-categories, and give some examples of classical results about low-degree Hochschild cohomology that carry over as expected and some that don't.

Vanessa Miemietz' slides

Toric degenerations provide a powerful link between the geometry of flag varieties and the combinatorics of convex polytopes. I will survey toric degenerations of flag varieties and Grassmannians arising from tropical geometry and representation theory. The comparison between degenerations coming from string polytopes and those produced by tropical cones will be a central theme, together with the interpretation of the resulting polytopes as Newton–Okounkov bodies for valuations associated to each tropical cone. I will also discuss a necessary condition for obtaining a toric initial ideal of the Grassmannian of 3-planes, and outline the main computational challenges this poses. This talk is based on joint works with Kris Shaw, and with Lara Bossinger, Sara Lamboglia, and Kalina Mincheva.

Fatemeh Mohammadi's slides (Updated)

Metrics on categories go back to the 1970s. In 2018 I proved that a certain completion of a triangulated category, with respect to a good metric, has a natural triangulated structure. The recent work shows two things: 1. It describes conditions on the metric, which guarantee that the construction is involutive. 2. It shows a compatibility with dg and stable-infinity enhancements. And this gives a powerful tool for proving new uniqueness-of-enhancement results.

Amnon Neeman's slides

In 2013 de la Peña initiated the systematic study of algebras of cyclotomic type, that is finite-dimensional algebras of finite global dimension such that some power of their Coxeter matrix is unipotent. For example, fractionally Calabi–Yau algebras have periodic Coxeter matrices. In this talk, we propose a class of algebras, which we call Serre cyclotomic, as a generalization of fractionally Calabi–Yau algebras and as a categorification of algebras of cyclotomic type. We study the dynamics of their Serre functors and discuss various examples.

Calvin Pfeifer's slides

Quasi-hereditary structures on monomial algebras are characterized completely by Green and Schroll, via a simple, concrete criterion. In this talk I will show that such quasi-hereditary algebras always admit an exact Borel subalgebra which can be described using similar techniques. Additionally, this exact Borel subalgebra gives rise to a Reedy decomposition in the sense of Dalezios and Šťovíček.

Anna Rodriguez Rasmussen's slides

We study 2-Calabi-Yau tilted algebras which are non-commutative Iwanaga Gorenstein algebras of Gorenstein dimension 1. In particular, we are interested in their syzygy categories or, equivalently, the stable categories of Cohen-Macauley modules CMP. First, we show that if an algebra A is Iwanaga-Gorenstein of Gorenstein dimension 1 then its stable category is generated under extensions by its radical rad A. Next, for a 2-Calabi-Yau tilted algebra A we provide an explicit relationship between the CMP category of A and its quotient A/AeA by an ideal generated by an idempotent e. Consequently, we obtain various equivalent characterizations of when the CMP category remains the same after passing to the quotient. This is joint work with Khrystyna Serhiyenko.

Ralf Schiffler's slides

In this talk we consider the problem of extensions of standard modules in highest weight categories or of Verma modules in Lie theory. This problem is surprisingly poorly understood. After a short motivation I will mention a general abstract approach to the problem and then illustrate concrete computation in an example. The talk is based on joint work with Jens Eberhardt.

Catharina Stroppel's slides

It is well-known that clusters in the \(A_n\) cluster algebra are in bijection with triangulations of a convex \((n + 3)\)-gon. In previous work, I showed a three-dimensional version of this result: namely, equivalence classes of maximal green sequences of linearly oriented \(A_n\) are in bijection with triangulations of a three-dimensional cyclic polytope with \(n + 3\) vertices. An obvious question to ask is: given a different orientation \(Q\) of the \(A_n\) Dynkin diagram, can we find a three-dimensional polytope whose three-dimensional triangulations are in bijection with equivalence classes of maximal green sequences of \(Q\)? In this talk, I will explain work that goes some way towards answering this question. Indeed, I show that for any cluster-tilting object \(T\) in the cluster category \(\mathcal{C}_n\) of type \(A_n\), there is an oriented matroid \(\mathcal{M}_{T}\) whose triangulations are in bijection with equivalence classes of maximal green sequences of \(\operatorname{End} T\). The definition of \(\mathcal{M}_{T}\) arises naturally from the extriangulated structure on \(\mathcal{C}_n\) which makes \(T\) projective.

Nicholas William's slides

This is a joint work with Federico Campanini and Francesca Fedele. In this short talk, we will introduce the lattice of pretorsion classes in a module category of a finite-dimensional algebra. We characterise and give a full classification of when this lattice is distributive, and further describe a connection to the famously known lattices of torsion classes.

Emine Yıldırım's slides

The homotopy category of a model structure on an additive category is calculated, which is known before only for the special case of exact model structures. Fibrantly weak factorization systems are introduced, fibrant model structures are constructed, and a one-one correspondence between fibrantly weak factorization systems and fibrant model structures is given. As applications we recover the \(\omega\)-model structures and the W-model structures, and their relations with exact model structures are discussed.

Pu Zhang' slides