Backgrounds of the Workshop

The Painlevé equations are a class of nonlinear ODEs that were discovered in the course of exploring special functions that generalize hypergeometric functions and elliptic functions. They are an important subject of study, as they are connected to various topics in mathematics and physics such as isomonodromic deformations, affine Weyl group symmetries, and the algebraic and symplectic geometric structures of their initial value spaces, matrix models, and so on. In these relationships, the τ-functions, which is analogous to θ-functions, play particularly significant roles.

Since the discovery in 2012 of an explicit formula for the asymptotic expansion of the τ-function of the Painlevé VI equation using conformal field theory, research aimed at investigating the structure of τ-functions associated with the general solutions of the Painlevé equations has become increasingly active. Progress has also been made in studying the τ-function using methods such as irregular conformal blocks, Nekrasov partition function, and topological recursion. In recent years, new approaches for studying the quantization of τ-functions have also been proposed.

However, many research challenges remain, including the comparison of τ-functions constructed using various approaches, their relationship with (wild) character varieties via the Riemann-Hilbert correspondence, connection problems, and more. The aim of this workshop is to promote discussions among researchers studying various aspects of the Painlevé equations by introducing the latest research results and unsolved problems, with the goal of achieving a deeper understanding of the properties of the τ-functions.

Timetable, Title and Abstract

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Financial Support

• RIMS, Kyoto University

• JSPS Grant-in-Aid for Scientific Research (A)   22H00094 (PI: Masa-Hiko Saito) 

• JSPS Grant-in-Aid for Challenging Research (Exploratory) 22K18669(PI: Masa-Hiko Saito)

• JSPS Grant-in-Aid for Challenging Research (Exploratory)   23K17654  (PI: Kohei Iwaki)

• JSPS Grant-in-Aid for Scientific Research (B)   24K00525 (PI: Kohei Iwaki)