Notes
Lectures
Cluster Categories and Related Topics
by Gordana Todorov
It was Fomin and Zelevinsky’s introduction of cluster algebras that inspired a large amount of interactions between different areas of mathematics, in particular researchers in representation theory and researchers in combinatorics started to work jointly on many problems (often related to cluster theory).
In this series of lectures, I will talk about:
1) Representations of Quivers (both without and with relations)
2) Auslander-Reiten theory
3) Cluster categories (acyclic case)
4) Relation between Cluster Algebras and Cluster Categories
5) Cluster theory and Friezes.
Introduction to Cluster Algebras and Combinatorics of Friezes
by Khrystyna Serhiyenko
A frieze is an infinite array of positive integers satisfying a certain diamond rule, that the determinant coming from the four neighboring entries equals 1. They were first introduced and studied in 1970’s by Conway and Coxeter, who proved a beautiful bijection between friezes and triangulations of polygons. The interest in friezes renewed with the introduction of Fomin and Zelevinsky’s cluster algebras in 2001, since cluster algebras of type A also correspond to triangulations of polygons.
In this course, we will explore the close connections between friezes and cluster algebras, as well as various generalizations of friezes inspired by these connections, such as sl_k friezes, infinite friezes, and tilings of the plane.
Schubert calculus and symmetric functions
by Maria Gillespie
How many lines pass through two points in the plane? How many points do a line and a conic intersect in? And given four lines drawn randomly in three-dimensional space, how many lines would we expect pass through all four of them?
The three questions above are classical problems in enumerative geometry, the study of counting the intersection points of geometric objects (like points, lines, planes, curves, and surfaces). We will lead participants on a tour through modern-day Schubert calculus, a combinatorial and geometric theory that allows us to answer many of these questions using the algebraic combinatorics of symmetric functions, permutations, and Young tableaux.
Tamari Lattices and Posets
by Emily Barnard
The Tamari lattice is a fundamental object of algebraic, enumerative and geometric combinatorics with connections to topology and physics. In this course, we begin by reviewing the connection between the Tamari lattice and Coxeter-Catalan combinatorics. We will then dive into its lattice-poset structure and explore the rich connection to torsion classes.
Talks
Cambrian Combinatorics on Quiver Representations (type A)
by Emily Gunawan
First, we will discuss a polygon model of the Auslander--Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. Next, we will introduce a new Catalan object which we call a maximal almost rigid representation. Finally, we will define a partial order on the set of maximal almost rigid representations and use our polygon model to show that this partial order is a Tamari or Cambrian lattice. This work is joint with Emily Barnard, Emily Meehan, and Ralf Schiffler.
Webs and tableau promotion
by Rebecca Patrias
Slides
In 2008, Petersen--Pylyavskyy--Rhoades proved that tableau promotion on 2-row and 3-row rectangular standard Young tableaux can be realized as rotation of certain planar graphs called webs, which were introduced by Kuperberg. In this talk, we will introduce webs, promotion and their result. After that, we discuss ongoing work to generalize further to the setting of K-theory combinatorics. This ongoing work is joint with Oliver Pechenik, Jessica Striker, and Juliana Tymoczko.
Properties of Random Walks through Characters and Young Tableaux
by Nadia Lafrènière
Slides
Random walks are a well-studied probabilistic object. We will look at discrete and finite versions of them. Most specifically, I will introduce random walks on the symmetric group, and show that we get a lot of information on them by studying their spectral properties, namely the eigenvalues and eigenvectors. These can be obtained with the help of representation theory. We will discuss the role of characters and Young tableaux in obtaining the eigenvalues.
The manifestations of Rowmotion in Representation Theory of Algebras
by Emine Yıldırım