Orthogonal polynomials and orthogonal polynomial ensembles

2024-12-25 ~ 2025-01-22

Orthogonal polynomials and orthogonal polynomial ensembles

Lecturer: Anton Nazarov (Visiting Professor)

Weekday: Wed, Fri

Time: 13:30-15:05

Venue: A3-3-201

Zoom: 518 868 7656

Password: BIMSA

Prerequisite

Basic notions of complex analysis, basic probability theory, linear algebra. Some knowledge of spectral theory and integrable systems can be useful but is not required. There is some overlap with material of my previous course "From free fermions to limit shapes and beyond".

Introduction

This minicourse presents a background for our joint research project with Anton Selemenchuk and Anton Dzhamay. In the lectures we will recall classical continuous and discrete orthogonal polynomials that admit hypergeometric representation and Askey scheme of their classification. We will discuss the connection of the orthogonal polynomials to determinantal ensembles in probability theory. Some asymptotic results, such as convergence to Tracy-Widom distribution will be considered. After reviewing classical picture we will move to semiclassical orthogonal polynomials that are obtained from classical ones by some weight modification. In particular, we will discuss Christoffel, Geronimus and Uvarov transformations. We will review our recent results with Anton Selemenchuk on the asymptotics of polynomials, obtained from Krawtchouk polynomials by Christoffel transformation. If time permits, we will also discuss Fredholm determinants and their connection to Painleve equations.

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.