Random matrices and asymptotic representation theory

Prerequisite

Some knowledge of representation theory of Lie groups or symmetric group and random matrix theory would be helpful. Demonstrations require the use of Python programming language and Sage computer algebra system, so some experience here would be a plus. There is some overlap with material of my previous course "From free fermions to limit shapes and beyond" and connections to the courses "Representation theory of symmetric groups", "Asymptotic representation theory" by Pavel Nikitin and "Topics in Random matrix theory" by Fan Yang.

Introduction

This short course is dedicated to connections between the theory of random matrices and asymptotic representation theory. In particular we will review the relation of Marchenco-Pastur distribution of eigenvalues limit density of Wishart random matrices to the limit shapes of Young diagrams for Schur-Weyl duality, derived by P. Biane.

Syllabus

1. Short introduction to random matrices: Gaussian Unitary Ensemble vs Wishart matrices.

2. Wigner semicircle law

3. Marchenco-Pastur law

4. Review of Young diagrams, Lie groups and symmetric group

5. Schur-Weyl duality

6. Plancherel measure and Wigner semicircle law

7. Limit shape of random Young diagrams from Schur-Weyl duality and Marchenco-Pastur distribution

8. Possible applications

Lecturer Intro

Anton Nazarov is an associate professor at Saint Petersburg State University, Russia. He completed his PhD at the department of high-energy and elementary particle physics of Saint Petersburg State University in 2012 under the supervision of Vladimir Lyakhovsky. In 2013-2014 he was a postdoc at the University of Chicago. Anton's research interests are representation theory of Lie algebras, conformal field theory, integrable systems, determinantal point processes.