Discrete Painlev\'e Equations

TL;DR

This article reviews the recent developments in discrete Painlevé equations, highlighting their connections to various mathematical fields and their role in understanding transcendental functions.

Contribution

It provides a comprehensive overview of the modern perspectives and interdisciplinary significance of discrete Painlevé equations over the past two decades.

Findings

Connections to random matrix theory, algebra, and geometry

Rich developments in transcendental solutions

Interdisciplinary insights into discrete integrable systems

Abstract

This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two decades have been rich and dynamic. These equations arise at the center of many viewpoints: random matrix theory, algebra, algebraic geometry, dynamical systems and the theory of transcendental functions. The purpose of this article is to reveal this confluence and modern perspectives on it.