Rep.Mutation.School 2022

# Mini-courses

## Hugh Thomas

Université du Québec à Montréal, Canada

## Research Talks

### Nathan Reading (North Carolina State University)

Title: Cluster scattering diagrams: Models and mutations

Abstract: Cluster algebras were invented/discovered in order to understand total positivity. But almost immediately, mathematicians started finding connections between the combinatorics/geometry/algebra of cluster algebras and other areas of mathematics, including representation theory. Scattering diagrams are a tool developed by geometers to understand mirror symmetry. They have been applied to prove key structural results about cluster algebras.

In this talk, I will introduce scattering diagrams and their theta functions, then introduce cluster algebras, and then describe the connection between the two by way of cluster scattering diagrams. I will motivate the need for models (combinatorial, rep-theoretic, etc.) for cluster scattering diagrams and survey some work in progress on these combinatorial models. I will also highlight the role of mutation in determining theta functions in some cases.

### Ralf Schiffler (University of Connecticut)

Title: Cluster algebras and knot theory

Abstract: To be posted!

### Hipolito Treffinger (Université de Paris)

Title: The wall-and-chamber structure as a tool

Abstract: In the course “Stability conditions in representation theory”, Sota Asai will introduce the wall-and-chamber structure of an algebra and he will explain some of its main properties. In this talk we will show how this geometric invariant can be used to solve important problems in representation theory, the so-called τ-Brauer-Thrall conjectures.

The classical Brauer-Thrall conjectures are two statements that relate the representation type of an algebra with the behaviour of the dimension vectors of its indecomposable representations. The recent developments in representation theory allowed the statement of two new conjectures relating the τ-tilting theory of an algebra and the behaviour of the dimension vectors of the bricks in its module categories. These are the τ-Brauer-Thrall conjectures.

The first τ-Brauer-Thrall conjecture has been shown by Schroll-T. and, independently, by Mousavand-Paquette. The first aim of this talk is to give a new proof of this conjecture using the wall-and-chamber structure of the algebra.

The second τ-Brauer-Thrall conjecture has been shown only for certain specific families of algebras in the work of Mousavand and Schroll-T.-Valdivieso, but it is open in general. We will finish the exposition by explaining an approach to the second τ-Brauer-Thrall conjecture using stability conditions, some of the evidence in favour of this approach and some of its problems.

## Prerequisites:

Participants are expected to be familiar with the basics of representation theory of quivers and some rudiments of

abelian categories and triangulated categories.

The following papers (and the references therein) can be used to suitably prepare for the materials discussed in this school.

For more background on representation theory of finite dimensional algebras, the following standard textbooks can be used: