Titles & abstracts

On the homotopy category of permutation modules over a finite group

This is ongoing joint work Martin Gallauer (MPI, Bonn). The homotopy category of permutation modules over a finite group in positive characteristic certainly fits the theme of the conference: It is perhaps one of the first triangulated categories we can consider in modular representation theory. And it has `beyond’ significance too, in connection with Artin motives in algebraic geometry and with cohomological Mackey functors in topology. Although very easy to define, we shall see that this tensor-triangulated category is remarkably rich. In particular it is in general too complicated to classify all its object up to isomorphism. However, its tensor-triangular geometry is reasonably understandable, as we shall explain.

Homotopical structures in representation theory

We will give an overview of recent developments in the homotopical representation theory of finite groups, with a particular focus on tt-geometric classification problems in mixed and intermediate characteristics. This is ongoing joint work with various subsets of Castellana, Gallauer, Heard, Naumann, Pol, and Sanders.

Green 2-functors

In the representation theory of finite groups, and beyond, many families of triangulated categories organise themselves into a "Mackey 2-functor": there is a category for each group, and there are restriction, induction and conjugation functors connecting them and satisfying suitable relations. This notion, a categorification of the classical Mackey functors of algebra and topology, was introduced and studied in joint work with Paul Balmer. In Nature, however, many interesting examples (derived and stable module categories of group algebras, equivariant stable homotopy categories, derived categories of equivariant sheaves, etc.) actually consist of tensor-triangulated categories, and should by all rights assemble into "Green 2-functors". In this talk I will define such a notion, drawing inspiration both from the classical theory of Green functors and from that of monoidal derivators. Green 2-functors are in fact ubiquitous in equivariant mathematics, and I will try to convince you of the usefulness of their theory by outlining some early applications.

Cluster structures for the \(A_{\infty}\) singularity

This talk is about a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules over the commutative ring \(\mathbb{C}[x,y]/(x^k)\). In the special case \(k=2\), \(\mathrm{Spec}(\mathbb{C}[x,y]/(x^2))\) is a type \(A_{\infty}\)-curve singularity and the ungraded version of our category has been studied by Buchweitz, Greuel, and Schreyer in the 1980s. We show that this Frobenius category has infinite type \(A\) cluster combinatorics, in particular, that it has cluster-tilting subcategories modelled by certain triangulations of the (completed) infinity-gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed infinity-gon. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

When aisles meet

Discrete cluster categories of type A are an exciting playing field on which to learn about infinite rank cluster combinatorics: On the one hand, they combinatorially behave, in many ways, in a familiar finite type A way. On the other hand, they exhibit new phenomena for which finite type A is "too small". One such phenomenon is the existence of non-trivial t-structures. In this talk, we describe the classification of t-structures in discrete cluster categories of type A via decorated non-crossing partitions and explain how they form a lattice under inclusion of aisles with meet given by intersection. If time permits, we will discuss the lattice of thick subcategories. This talk is based on joint work with Alexandra Zvonareva.

Representations and fibres

The talk will be about the representation theory of a finite projective algebra \(A\), over a commutative noetherian ring \(R\). It will address the question: How much of the representation theory of \(A\) can be understood in terms of the representation theory of the fibres of \(A\) over \(\rm{Spec} R\)? The talk is based on joint work with (subsets of) Benson, Gnedin, Krause, and Pevtsova.

The Triangulated Auslander–Iyama Correspondence (2 talks)

The purpose of these talks is to present a result that establishes the uniqueness of (DG) enhancements for algebraic triangulated categories which admit a generator with a strong regularity property (a so-called \(d \mathbb{Z}\)-cluster tilting object). Moreover, our result classifies such categories in terms of minimal algebraic data: a twisted periodic Frobenius algebra equipped with an automorphism. We will discuss some aspects of the proof as well as applications of our main results (see also B. Keller's talk).

The \(Q\)-shaped derived category

(Report on joint work with Henrik Holm.) Let \(Q\) be the quiver \(A^{\infty}_{\infty}\) with the relations \(d^2 = 0\); that is, any two consecutive arrows compose to zero. If \(R\) is a ring, then a representation of \(Q\) with values in \(\rm{Mod}(R)\) is a chain complex of \(R\)-modules.

Hence, classic homological algebra can be viewed as the study of representations of \(Q\). It turns out that several aspects of the theory work because \(Q\) is self-injective. Replacing \(Q\) by another self-injective quiver with relations permits the theory to be generalised. There is considerable scope for such replacements; among the possibilities are cyclic and linear quivers with the relations \(d^N = 0\).

The generalised setup permits a notion of quasi-isomorphism, which enables the definition of a "\(Q\)-shaped derived category". Computations in this category are facilitated by two underlying model category structures, which generalise the classic projective and injective model category structures on chain complexes.

An introduction to the Donovan-Wemyss conjecture

We will give a historical introduction to Donovan-Wemyss' conjecture stating that a compound Du Val singularity is determined by the derived equivalence class of the contraction algebra associated with any crepant resolution (2013). More and more evidence for the conjecture has been accumulated in work by Donovan-Wemyss, Wemyss, Toda, Hua-Toda, Hua, August, Hua-K and others. We will conclude by explaining how the conjecture follows by combining the work of August and Hua-K with the Auslander-Iyama correspondence for triangulated categories, a recent theorem due to Gustavo Jasso and Fernando Muro.

Nakayama functors and monomorphism categories

The notion of a Serre functor for triangulated categories as introduced by Bondal and Kapranov is now widely used. For derived categories of algebras, it is given by the derived Nakayama functor, and its importance in representation theory goes back to the fundamental work of Auslander and Reiten on almost split sequences. In this talk, we will study Nakayama functors, and a relative notion due to Kvamme, in the abelian setting. We will discuss some of their properties that are analogous to Serre functors, and give applications to the study of monomorphism categories of quivers in rings, vastly generalising work of Ringel and Schmidmeier. This is based on joint work with Nan Gao, Sondre Kvamme, and Chrysostomos Psaroudakis.

The spectrum of a well-generated tensor triangulated category

In many tensor triangulated categories the thick subcategories of the compact objects and the localizing subcategories of the whole category are classified by topological spaces which have the same underlying set. Examples include the derived category of a commutative noetherian ring and the stable module category of a finite group. A well-generated triangulated category is generated by a set of \(\alpha\)-compact objects for some regular cardinal \(\alpha\). Under some mild conditions the \(\alpha\)-localizing subcategories of \(\alpha\)- compact objects are also classified by a topological space. In my talk I explain the connection between the topological spaces classifying the thick subcategories of the compact objects, the \(\alpha\)-localizing subcategories of the \(\alpha\)-compact objects, and the localizing subcategories. This is joint work with Henning Krause.

On some invariants of stable module categories of finite group algebras

There are two tensor triangulated categories of interest in the context of finite group algebras - the stable module category, and the stable category of bimodules which are finitely generated projective as left and right modules. The former is symmetric monoidal, embeds in the latter which is not symmetric, and both are invariant as triangulated categories under stable equivalences of Morita type. We consider invariants of both of a different nature but with the same goal of capturing numerical invariants. For the stable module category we show that although this category does not have any nontrivial t-structures, it does have distinguished abelian subcategories which capture some numerical invariants. Hochschild cohomology in positive degree is an invariant of the second category under consideration, and we present some dimension calculations for symmetric group algebras - this part is joint work with Dave Benson and Radha Kessar.

Vanishing K-theory and bounded t-structures

We will begin with a quick reminder of algebraic K-theory, and some classical results. The talk will then focus on a striking 2019 article by Antieau Gepner and Heller - it turns out that there are K-theoretic obstructions to the existence of bounded t-structures.

The result suggests many questions. A few have already been answered, but many remain open. We will concentrate on the many possible directions for future research.

Finite approximations as a tool for studying triangulated categories

A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.

And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions.

And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.

And what makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a generalization of a theorem of Rouquier's, a short, sweet proof of Serre's GAGA theorem and a proof of a conjecture by Antieau, Gepner and Heller.

Some applications of extriangulated categories

In representation theory of finite dimensional algebras, extension-closed full subcategories of triangulated categories seem to arise naturally in various contexts, such as additive categorification of cluster algebras, or tau-tilting and silting theory. In order to axiomatize those subcategories, the notion of an extriangulated category was introduced in collaboration with Hiroyuki Nakaoka. In this talk, we will motivate the definition of extriangulated categories, and illustrate their use with several applications in homotopical algebra, representation theory and combinatorics. The talk is based on collaborations with Hiroyuki Nakaoka, Arnau Padrol, Vincent Pilaud and Pierre-Guy Plamondon.

Local to global approach to support theories and \(1/2\) quantum flag variety

I’ll discuss support theories for various classes of finite dimensional Hopf algebras (over a field). The emphasis will be on how the support data for representations of a more complex algebraic structure gets assembled from the "local" information about the representation. We’ll start with the classical example of Quillen stratification theorem for finite groups, continue with p-Lie algebras and finite group schemes and their supersized versions and move to small quantum groups where the half quantum flag variety makes an appearance. Joint work with D. Benson, S. Iyengar and H. Krause; and with C. Negron.

Generating the derived category

The unbounded derived category of (right) modules over a ring is a triangulated category with infinite products and coproducts. As a triangulated category with coproducts it is easy to see that it is generated by the projective modules, and similarly it is generated as a triangulated category with products by the injective modules.

I will discuss the question of whether it is generated as a triangulated category with coproducts by the injective modules, or as a triangulated category with products by the projective (or flat) modules. I will describe the relationship with the finitistic dimension conjecture, as well as some more recent results.

Many examples of non-Fourier-Mukai functors

The first example of a non-Fourier-Mukai functor between particularly nice triangulated categories – the derived categories of smooth projective varieties – was given by myself, Van den Bergh and Neeman in 2015. I will show that this is not a pathological example by providing a way to construct a non-Fourier-Mukai functor from the derived category of any smooth projective variety of dimension greater or equal to 3 admitting a tilting bundle. This is joint work with Theo Raedschelders and Michel Van den Bergh.

Perverse equivalences and applications

I will discuss the theory of perverse equivalences introduced jointly with Joe Chuang. I will focus on certain combinatorial aspects and some relations with decomposition matrices. This brings in Hilbert schemes (joint with Olivier Dudas), quantum cohomology (after Bezrukavnikov and Okounkov) and cactuses (after Losev).

Some remarks on lattices of thick subcategories

I'll discuss some recent thoughts that Sira Gratz and I have had on the problem of understanding the lattice of thick subcategories of an essentially small triangulated category. In particular, I'll briefly cover what one gets from distributivity of this lattice and what we might try to do without it.