"Random Young diagrams and tableaux"

Young diagrams and (standard) Young tableaux are central objects in the theories of symmetric functions and of symmetric group representations. In this lecture, we take a probabilistic viewpoint on these objects. What does a large random Youg diagram or Young tableau, taken with an appropriate probability distribution, look like? We will discuss three sets of tools which have been used to attack these questions: the entropy method, the representation theoretical approach, and (briefly) the determinantal point process approach.

Lecture notes and exercises at this link

"q-counting and invariant theory"

We hope to illustrate some interesting counting and q-counting formulas, with explanations from representation theory and/or invariant theory of reflection groups. As a reference, see this chapter based on lectures at ECCO 2018

Lecture notes

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Lecture 5

Exercises are here

"The ubiquity of crystal bases"

This course provides an introduction to crystal bases as a combinatorial tool to study representation theory. We will discuss Kashiwara crystals, tensor products of crystals, Stembridge crystals, virtual crystals, crystals as a tool to study Schur expansions of symmetric functions, and if time allows a crystal for Stanley symmetric functions. We will also discuss ways to explore crystals using SageMath. As references, you may use my book with Dan Bump "Crystal bases: Representations and combinatorics" and the online tutorial https://doc.sagemath.org/html/en/thematic_tutorials/lie.html.

Exercises 1

Exercises 2

Lecture 5