Program, Titles and Abstracts

Title :  From Elliptic curve to Painleve equations via WKB and topological recursion

Abstract :  I’ll explain how the tau-function (associated with the general solution) for the first Painlevé equation is constructed as a discrete Fourier transform of the topological recursion partiton function.  This can be regarded as an irregular singular analogue of the Kyiv formula.  If time permits, I’ll also explain about an expected relation between topological recursion and irregular conformal blocks (on-going joint work with N. Iorgov, O, Lisovyy and Y. Zhuravlev).   

Title :  A unified family of PJ-hierarchies (J=I,II,IV,34) with a large parameter

Abstract :   In this talk, we consider the common structures among PJ-hierarchies(J=I,II,IV,34) with a large parameter from a view point of the exact WKB analysis. 

In the first part, we present results related to the construction of instanton-type solutions with sufficiently many free parameters which can describe the Stokes phenomenon of the Borel sum of a 0-parameter solution. 

In the second part, we discuss the Stokes geometry for a unified family of PJ-hierarchies(J=I,II,IV,34) with a large parameter. We study the relation between the Stokes geometry of the system of non-linear ordinary differential equations and that of its underlying Lax pair. 

Title :  Geometric aspects of delay-differential Painlevé equations

Abstract :   In this talk we discuss several geometric aspects of some ordinary delay-differential equations that can be regarded as analogues of Painlevé equations. These are in a sense a hybrid of differential and discrete, involving shifts and derivatives with respect to a single independent variable, but the theory of Painlevé-type equations in this class is in its infancy compared to the purely discrete and differential cases. We begin by discussing a kind of singularity confinement phenomenon in the delay-differential setting, which turns out to be more complicated and richer than in the discrete case where it plays a key role in the geometric theory of discrete Painlevé equations. We will then present some results on a delay-differential analogue of the first Painlevé equation, where formal entire solutions are constructed in terms of certain polynomials that share properties with the classical Bernoulli and Euler polynomials and turn out to be related to the problem of calculation of the Masur-Veech volumes of moduli spaces of quadratic differentials.

Part of this talk is based on joint work with J. Gibbons and A. P. Veselov.

Title:  Twisted polar-parts varieties

Abstract :   In this talk, I will introduce a twisted version of Boalch’s polar-parts manifolds, which is an additive analogue of twisted wild character varieties. A Twisted polar-parts variety parameterizes meromorphic connections on the trivial principal bundle with a reductive structure group satisfying a certain equivariance condition for actions of a finite group on the structure group and the base space. I will also give some examples of such connections using the Fourier-Laplace transform.

Title :  Oper and Canonical coordinates of the moduli spaces of parabolic connections and parabolic Higgs bundles

Abstract :  In this talk, I will explain how we can itroduce  a system of coordinates on a dense Zarisiki open set of the moduli spaces of parabolic connections and parabolic Higgs bundles by using Oper and apparent singularity.  In rank 2 case, we can showed that these coordinates give  canonical coordinates with respect to  the natural algebraic symplectic structure introduced by Inaba-Iwasaki-Saito.  We will also explain the situation of higher rank case.  If time allow, we will  explain about the geometric structure of the moduli spaces by using these coordinates.  This talk is based on  joint works with Arata Komyo, Frank Loray and Szilard Szabo arXiv:2309.05012. 2023 and Szilard Szabo.