Program

Schedule

Each talk has a duration of 50 minutes; there will be a 10 minutes break between each talk. You can navigate the abstracts by clicking on the green button on the bottom-right corner of the screen labelled “Table of contents”.

Times are displayed in your local time zone.

First week (November 9-13)

Monday, November 9, 2020 4:00 PM - Pavel Etingof L1
Monday, November 9, 2020 5:10 PM - Srikanth Iyengar L1

Tuesday, November 10, 2020 8:00 AM - Bernhard Keller L1
Tuesday, November 10, 2020 9:10 AM - Magnus Botnan L1

Wednesday, November 11, 2020 8:00 AM - Bernhard Keller L2
Wednesday, November 11, 2020 9:10 AM - Magnus Botnan L2

Friday, November 13, 2020 8:00 AM - Bernhard Keller L3
Friday, November 13, 2020 9:10 AM - Magnus Botnan L1

Second week (November 16-20)

Monday, November 16, 2020 4:00 PM - Pavel Etingof L2
Monday, November 16, 2020 5:10 PM - Srikanth Iyengar L2

Tuesday, November 17, 2020 4:00 PM - Pavel Etingof L3
Tuesday, November 17, 2020 5:10 PM - Srikanth Iyengar L3

Wednesday, November 18, 2020 8:00 AM - Hiroyuki Minamoto
Wednesday, November 18, 2020 9:10 AM - Sibylle Schroll L1

Friday, November 20, 2020 4:00 PM = Sarah Witherspoon
Friday, November 20, 2020 5:10 PM - Paul Balmer

Third week (November 23-25)

Monday, November 23, 2020 8:00 AM - Sota Asai
Monday, November 23, 2020 9:10 AM - Haruhisa Enomoto
Monday, November 23, 2020 4:00 PM - Conference talk?
Monday, November 23, 2020 5:10 PM - Conference talk?

Tuesday, November 24, 2020 8:00 AM - Fan Qin
Tuesday, November 24, 2020 9:10 AM - Sibylle Schroll L2
Tuesday, November 24, 2020 4:00 PM - Sarah Scherotzke
Tuesday, November 24, 2020 5:10 PM - Steven Sam

Wednesday, November 25, 2020 8:00 AM - Conference talk?
Wednesday, November 25, 2020 9:10 AM - Sibylle Schroll L3
Wednesday, November 25, 2020 4:00 PM - Sergey Fomin
Wednesday, November 25, 2020 5:10 PM - ICRA Award Ceremony and Quiz

Abstracts

Workshop

Magnus Botnan (Vrije Universiteit Amsterdam)

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).

Pavel Etingof (Massachusetts Institute of Technology)

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.

Srikanth Iyengar (University of Utah)

Duality for Gorenstein algebras

Bernhard Keller (Université de Paris)

An introduction to relative Calabi-Yau structures

Sibylle Schroll (University of Leicester)

Recent developments in gentle algebras

Conference

Sota Asai (Osaka University)

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 Br"{u}stle–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.

Paul Balmer (University of California, Los Angeles)

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.

Haruhisa Enomoto (Nagoya University)

ICE-closed subcategories and wide \tau-tilting modules

Sergey Fomin (University of Michigan)

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.

Hiroyuki Minamoto (Osaka Prefecture University)

Quiver Heisenberg algebras: a cubical analogue of preprojective algebras

Fan Qin (Shanghai Jiao Tong University)

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.

Steven Sam (University of California, San Diego)

Curried Lie algebras

Sarah Scherotzke (Université du Luxembourg)

TBD

Sarah Witherspoon (Texas A&M University)

Varieties for Representations and Tensor Categories