Talks and abstracts

Quiver Representations in Topological Data Analysis

The goal of these three lectures is to highlight the role of quiver representations in the field of topological data analysis (TDA). Emphasis will be put on the interplay between the pure and applied. Familiarity with simplicial (co-)homology will be assumed.

Lecture 1: Persistent homology in a single parameter

Persistent homology is a central topic in the burgeoning field of topological data analysis. The key idea is to study topological spaces constructed from data and infer the ‘‘shape’’ of the data from topological invariants. The term ‘’persistent’’ refers to the fact that the construction of these spaces usually depends on one or more parameters, and in order to obtain information about the data in a stable and robust way, it is crucial to consider how the family of resulting invariants relate across scales. This naturally leads to a representation of a totally ordered set.

In this first lecture I will motivative persistent homology in a single parameter, introduce the necessary terminology, and state foundational results.

Lecture 2: Multiparameter persistent homology part 1

Multiparameter persistent homology is a vibrant subfield of topological data analysis which has attracted much attention in recent years. It has become evident that the transition from a single to multiple parameters comes with significant computational and mathematical challenges. At the level of representation theory, this can be understood by the fact that one is studying representations of a partially ordered set of wild representation type.

In this lecture we shall identify settings for which the theory in the first lecture generalizes to more general posets. Of particular interest is level-set zigzag persistent homology.

Lecture 3: Multiparameter persistent homology part 2

In this lecture we will consider models for constructing representations of posets for which most of the theory developed in the first lecture does not generalize in a reasonable way. However, we shall see that we still can extract useful invariants for the purpose of data analysis. Our primary motivation will come from clustering (in the data-scientific sense).

Symmetric tensor categories

Lecture 1: Algebra and representation theory without vector spaces.

A modern view of representation theory is that it is a study not just of individual representations (say, finite dimensional representations of an affine group or, more generally, supergroup scheme \(G\) over an algebraically closed field \(k\)) but also of the category \(\operatorname{Rep}(G)\) formed by them. The properties of \(\operatorname{Rep}(G)\) can be summarized by saying that it is a symmetric tensor category (shortly, STC) which uniquely determines \(G\). A STC is a natural home for studying any kind of linear algebraic structures (commutative algebras, Lie algebras, Hopf algebras, modules over them, etc.); for instance, doing so in \(\operatorname{Rep}(G)\) amounts to studying such structures with a \(G\)-symmetry. It is therefore natural to ask: does the study of STC reduce to group representation theory, or is it more general? In other words, do there exist STC other than \(\operatorname{Rep}(G)\)? If so, this would be interesting, since algebra in such STC would be a new kind of algebra, one “without vector spaces”. Luckily, the answer turns out to be “yes”. I will discuss examples in characteristic zero and \(p>0\), and also Deligne’s theorem, which puts restrictions on the kind of examples one can have.

Lecture 2: Representation theory in non-integral rank.

Examples of symmetric tensor categories over complex numbers which are not representation categories of supergroups were given by Deligne-Milne in 1981. These very interesting categories are interpolations of representation categories of classical groups \(\operatorname{GL}(n)\), \(\operatorname{O}(n)\), \(\operatorname{Sp}(n)\) to arbitrary complex values of \(n\). Deligne later generalized them to symmetric groups and also to characteristic \(p\), where, somewhat unexpectedly, one needs to interpolate \(n\) to \(p\)-adic integer values rather than elements of the ground field. I will review some of the recent results on these categories and discuss algebra and representation theory in them.

Lecture 3. Symmetric tensor categories of moderate growth and modular representation theory.

Deligne categories discussed in Lecture 2 violate an obvious necessary condition for a symmetric tensor category (STC) to have any realization by finite dimensional vector spaces (and in particular to be of the form \(\operatorname{Rep}(G)\)): for each object \(X\) the length of the \(n\)-th tensor power of \(X\) grows at most exponentially with \(n\). We call this property “moderate growth”. So it is natural to ask if there exist STC of moderate growth other than \(\operatorname{Rep}(G)\). In characteristic zero, the negative answer is given by the remarkable theorem of Deligne (2002), discussed in Lecture 1. Namely Deligne’s theorem says that a STC of moderate growth can always be realized in supervector spaces. However, in characteristic \(p\) the situation is much more interesting. Namely, Deligne’s theorem is known to fail in any characteristic \(p>0\). The simplest exotic symmetric tensor category of moderate growth (i.e., not of the form \(\operatorname{Rep}(G)\)) for \(p>3\) is the semisimplification of the category of representations of \(\mathbb{Z}/p\), called the Verlinde category. For example, for \(p=5\), this category has an object \(X\) such that \(X^2=X+1\), so X cannot be realized by a vector space (as its dimension would have to equal the golden ratio). I will discuss some aspects of algebra in these categories, in particular failure of the PBW theorem for Lie algebras (and how to fix it) and Ostrik’s generalization of Deligne’s theorem in characteristic \(p\). I will also discuss a family of non-semisimple exotic categories in characteristic \(p\) constructed in my joint work with Dave Benson and Victor Ostrik, and their relation to the representation theory of groups \((\mathbb{Z}/p)^n\) over a field of characteristic \(p\).

Duality for Gorenstein algebras

Lecture I: Gorenstein algebras

The overarching theme is dualities (note the plural) for Gorenstein algebras. The plan for the first lecture is to introduce a class of Gorenstein algebras wide to cover the main examples that come up in commutative algebra and in representation theory. I will present some examples, and then discuss the duality theorems of Auslander and Grothendieck. I will also introduce the class of Maximal Cohen-Macaulay modules, and Gorestein projectives modules. If time permits, I will discuss abstract Serre duality.

Lecture II: Local cohomology for algebras finite over their center

In this lecture I will introduce the various (triangulated) categories of modules over Gorenstein algebras. One highlight will be Buchweitz’s theorem for this class of algebras. Next I will discuss the theory of local cohomology and support for these categories, from a point of view developed with Benson, Krause, and myself. To that end, I will recall some basic facts on injective modules, from Gabriel’s thesis.

Lecture III: Local duality for Gorenstein algebras

In this lecture I will discuss the Nakayama functor on the module category, and its extension to various categories introduced in Lecture II. The goal will be to present a local duality theorem for G-projectives over Gorenstein algebras that encompasses Grothendieck duality (for commutative rings) and Auslander duality (for finite dimensional algebras).

An introduction to relative Calabi-Yau structures

Following Ginzburg, an n-Calabi-Yau structure on a smooth (dg) algebra is an isomorphism between the identity bimodule and its bimodule dual shifted by n degrees to the left. Such structures should be thought of as noncommutative analogues of orientations on n-dimensional manifolds. Similarly, relative Calabi-Yau structures should be thought of as noncommutative analogues of orientations on n-dimensional manifolds with boundary. Here, the orientation should be considered as a datum attached to the inclusion of the boundary.

The history of relative Calabi-Yau structures is short: They were first mentioned in a 90-page survey paper by Bertrand Toen in 2014. They were systematically developed and investigated by Brav and Dyckerhoff in two papers (preprints in 2016 and 2018). One of their main achievements is a glueing construction analogous to the glueing of manifolds with boundary. As an application, they constructed Calabi-Yau structures on topological Fukaya categories of framed punctured Riemann surfaces. Wai-kit Yeung introduced (deformed) relative Calabi-Yau completions in a preprint in 2016. These can be viewed as far-reaching generalizations of (deformed) preprojective algebras. As their name suggests, they carry canonical relative Calabi-Yau structures. An important subclass is given by relative Ginzburg algebras of ice quivers with potential. Bozec-Calaque-Scherotzke recently showed (06/2020), that (non deformed) relative CY-completions are noncommutative analogues of conormal bundles. In particular, absolute Calabi-Yau completions (2011) are noncommutative cotangent bundles.

The importance of (absolute) 2- and 3-Calabi-Yau structures for the theory of cluster algebras (without coefficients or with “Frobenius coefficients”) is well-known. Work of Pressland and others strongly suggests that relative 2- and 3-Calabi-Yau structures should play a similarly important role in the deeper understanding of cluster algebras with coefficients. Ongoing work by Dyckerhoff and by Merlin Christ will likely show this for the cluster algebras with coefficients associated to marked surfaces with boundary. One can also expect that (higher-dimensional) relative Calabi-Yau structures will be useful in higher Auslander-Reiten theory. Again, this is suggested by work of Pressland and by the recent observation that the relative inverse Serre functor is equivalent to the higher Auslander-Reiten translation.

The material of these lectures is mostly taken from Brav-Dyckerhoff’s first article (Compositio 2019), the 2016 preprint by Yeung, work of Pressland and the ongoing Ph. D. thesis of Yilin Wu. We will start by recalling the necessary facts on differential graded (=dg) categories, their derived categories and derived functors. We will then present (left and right, absolute) Calabi-Yau structures and Calabi-Yau completions. We will introduce relative Calabi-Yau completions and illustrate them on many examples. This will also be the opportunity to establish the link between the relative inverse Serre functor and the higher Auslander-Reiten translation. We will then recall the necessary facts on Hochschild and cyclic homology and introduce (left and right) relative Calabi-Yau structures. We will see how relative Ginzburg algebras arise as deformed relative Calabi-Yau completions. This will entail strong homological properties for the relative Jacobian algebras obtained as zero-homologies of relative Ginzburg algebras concentrated in degree 0. It will also lead to a natural proof of the derived equivalence between the relative Ginzburg algebras associated to ice quivers with potential related by Pressland’s construction of mutation. We will end with an introduction to Brav-Dyckerhoff’s theorem on the composition of Calabi-Yau cospans.

Recent developments in gentle algebras

Gentle algebras are a class of quadratic monomial algebras given by quiver and relations originating in the work of Assem and Skowroński on iterated tilted algebras of affine type A in the 1980s. Since their inception, there has been a constant interest and study of their representation theory. Many aspects of their module categories and also of their derived categories are well understood. They are special biserial, hence of tame representation type and their representation theory is governed by the so-called string combinatorics, that is certain words in the alphabet given by the arrows and their formal inverses. Remarkably, the string and band combinatorics also controls their derived categories.

In the last 10 years there has been a renewed and increasing interest in gentle algebras, in particular in relation to geometric surface models connecting gentle algebras with other areas of mathematics. The relation of gentle algebras with marked surfaces first arises in the context of cluster theory and its connection to triangulations of surfaces. Building on work by Caldero-Chapoton-Schiffler (2006) for cluster-tilted algebras of type A, Assem-Brüstle-Charbonneau-Jodoin-Plamondon (2010) show that the Jacobian algebra of a quiver with potential associated to a triangulation of an oriented surface with marked points in the boundary is a gentle algebra and that the homotopy classes of curves in the surface parametrise the indecomposable modules over the Jacobian algebra. Much interest and work on the module category of gentle algebras and related geometric models ensued.

The origin of geometric models for the derived category of gentle algebras lies in the work of Haiden-Katzarkov-Kontsevich (2017) where they relate the partially wrapped Fukaya category of an oriented surface with stops to the derived category of a (graded) gentle algebra of finite global dimension viewed as differential graded algebra with zero differential. Lekili and Polishchuk (2020) then show that every graded gentle algebra of finite global dimension gives rise to a partially wrapped Fukaya category of an oriented surface with stops.

In these lectures we will give an overview of some of the geometric models for gentle algebras arising in representation theory related to the surface constructions coming from cluster theory on the one hand and related to Fukaya categories on the other. We will then survey some of the very recent representation theoretic developments in gentle algebras in connection with the geometric models for their module and derived categories.

The wall-chamber structures of the real Grothendieck groups

For a given finite-dimensional algebra \(A\) over a field, stability conditions introduced by King define the wall-chamber structure of the real Grothendieck group \(K_0(\operatorname{proj} A)_\mathbb{R}\), as in the works of Brüstle–Smith–Treffinger and Bridgeland. In this talk, I would like to explain my result that the chambers in this wall-chamber structure are precisely the open cones associated to the basic 2-term silting objects in the perfect derived category. As one of the key steps, I introduced an equivalence relation called TF equivalence by using numerical torsion pairs of Baumann–Kamnitzer–Tingley. If time permits, I will give some further results which were obtained in the ongoing joint work with Osamu Iyama.

Derived category of permutation modules

The general theme of this joint work with Martin Gallauer is the study of how much of representation theory of a finite group is controlled by permutation modules. I shall recall basic definitions and state our result about finite resolutions by p-permutation modules in positive characteristic p. This is related to a reformulation in terms of derived categories. Time permitting, I shall discuss coefficients in more general rings than fields. This will relate to the singularity category of such rings, as constructed by Krause.

ICE-closed subcategories and wide \(\tau\)-tilting modules

I’ll introduce ICE-closed subcategories of an abelian length category, subcategories closed under taking Images, Cokernels and Extensions. Both torsion classes and wide subcategories of an abelian category are ICE-closed. We will see that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For a finite-dimensional algebra, I’ll introduce the notion of wide \(\tau\)-tilting modules, and extend Adachi-Iyama-Reiten’s bijection between support \(\tau\)-tilting modules and torsion classes to a bijection between wide \(\tau\)-tilting modules and ICE-closed subcategories. If time permits, I’ll talk about some results on ICE-closed subcategories over hereditary algebras and Nakayama algebras. This talk is based on a joint work with Arashi Sakai (Nagoya University).

Expressive curves

We call a real plane algebraic curve C expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject to a mild technical condition), describe several constructions that produce expressive curves, and relate their study to the combinatorics of plabic graphs, their quivers and links. This is joint work with E. Shustin.

Quiver Heisenberg algebras: a cubical analogue of preprojective algebras

In this abstract \(K\) denotes a field of \(\mathrm{char } K = 0\) and \(Q\) denotes a finite acyclic quiver.

Recall that the preprojective algebra \(\Pi(Q) = K\overline{Q}/(\rho )\) is the path algebra \(K \overline{Q}\) of the double quiver \(\overline{Q}\) of \(Q\) with the mesh relation \(\rho=\sum_{\alpha \in Q_{1}} \alpha \alpha^{*} - \alpha^{*} \alpha\). It is an important mathematical object having rich representation theory and plenty of applications. In this joint work with M. Herschend, we study a central extension \(\Lambda(Q) := K \overline{Q}/([a, \rho] \mid a \in \overline{Q}_{1})\) of \(\Pi(Q)\) (where \([-,+]\) is the commutator) under the name ``quiver Heisenberg algebras’’. We note that our algebra \(\Lambda(Q)\) is a special case of central extensions \(\Lambda(Q)_{\lambda, \mu}\) of the preprojective algebras introduced by Etingof-Rains, which is a special case of algebras \(\Lambda(Q)_{P}\) introduced by Cachazo-Katz-Vafa, which is a special family of the deformation of preprojective algebras introduced by Crawley-Boevey-Holland. However, our algebra \(\Lambda(Q)\) of very special case has intriguing properties, among other things it provides an exact sequence of \(K Q\)-bimodules which can be called the universal Auslander-Reiten triangle.

We may equip \(\Lambda\) with a grading by setting \(\mathrm{deg} \ \alpha = 0, \mathrm{deg} \ \alpha^{*} : =1\) for \(\alpha \in Q_{1}\). We introduce an algebra to be \(B(Q) := {\tiny \begin{pmatrix} K Q & \Lambda_{1} \\ 0 & K Q \end{pmatrix}}\) where \(\Lambda_{1}\) is the degree \(1\)-part of \(\Lambda(Q)\). Together with previous results about \(\Lambda(Q)\) by Etingof-Rains, Etingof-Latour-Rains and Eu-Schedler, our results show that \(\Lambda(Q)\) and \(B(Q)\) posses properties that, can be looked as one-dimension higher version of that of the preprojective algebras \(\Pi(Q)\) and the path algebras \(K Q\).

Bases of cluster algebras

One of Fomin and Zelevinsky’s main motivations for cluster algebras was to study the dual canonical bases. Correspondingly, it had been long conjectured that the quantum cluster monomials (certain monomials of generators) belong to the dual canonical bases up to scalar multiples. Geiss-Leclerc-Schröer proved an analogous statement that the cluster monomials belong to the dual semi-canonical bases, which are examples of generic bases.

In a geometric framework for cluster algebras, Fock and Goncharov expected that cluster algebras possess bases with good tropical properties.

In this talk, we consider a large class of quantum cluster algebras called injective-reachable (equivalently, there exists a green to red sequence). We study their tropical properties and obtain the existence of generic bases. Then we introduce the (common) triangular bases, which are Kazhdan-Lusztig type bases with good tropical properties. We verify the above motivational conjecture in full generality and, by similar arguments, a conjecture by Hernandez-Leclerc about monoidal categorification.

Curried Lie algebras

A representation of \(gl(V)\) is a map \(V \otimes V^* \otimes M \to M\) satisfying some conditions, or via currying, it is a map \(V \otimes M \to V \otimes M\) satisfying different conditions. The latter formulation can be used in more general symmetric tensor categories where duals may not exist, such as the category of sequences of symmetric group representations under the induction product. Several other families of Lie algebras have such “curried” descriptions and their categories of representations have nice compact descriptions as representations of the (walled) Brauer category, partition category, and variants. I will explain a few examples in detail and how we came to this definition. This is joint work with Andrew Snowden.

Cotangent complexes of moduli spaces and Ginzburg dg algebras

First, we give an introduction to the notion of moduli stack of a dg category. Then we explain what shifted symplectic structures are and how they are connected to Calabi-Yau structures on dg categories. More concretely, we will show that the cotangent complex to the moduli stack of a dg category A is isomorphic to the moduli stack of the Calabi-Yau completion of A, answering a conjecture of Keller-Yeung. This is joint work with Damien Calaque and Tristan Bozec arxiv.org/abs/2006.01069

t-structures and co/tilting theory via Grothendieck derivators

During this talk we show how (strong and stable) Grothendieck derivators can be used to study t-structures in their underlying triangulated categories. In particular, given a strong and stable derivator D, we show that t-structures on the underlying category D(1) correspond bijectively with suitably defined “homotopy factorization systems” on D. As an immediate consequence, one obtains a nice criterion to decide if the heart of a given t-structure is a Grothendieck category.

In the second part of the talk we describe a general construction of realization functors. In particular, given a t-structure t=(U,ΣV) on D(1), and letting A=U∩ΣV be its heart, we construct a morphism of prederivators
real : Der(A)→D where Der(A) is the natural prederivator enhancing the (unbounded) derived category of A. Furthermore, we give criteria for this morphism to be fully faithful and essentially surjective. If the t-structure t is induced by a suitably “bounded” co/tilting object, real is always an equivalence. Our construction unifies and extends most of the derived co/tilting equivalences appeared in the literature in the last years.

Varieties for Representations and Tensor Categories

Algebraic geometry is used in representation theory to uncover information through the assignment of support varieties to modules. This theory began with finite group representations, and has been generalized in many directions. In this talk we will introduce the general theory in the contexts of finite dimensional Hopf algebras and finite tensor categories. These include representations of finite group algebras, restricted Lie algebras, and small quantum groups. We will discuss applications and recent developments.