Workshop on:

Isomonodromic deformations and related topics

Organisers: Martin Guest (Tokyo Metropolitan University) and Masa-Hiko Saito (Kobe University)

Supported by JSPS Grant-in-Aid for Scientific Research (S)19104002 (PI: Masa-Hiko Saito) and (A) 21244004 (PI: Martin Guest)

PROGRAM

Friday 27 January

10:00-11:00 Masa-Hiko Saito (Kobe University), "Geometry of moduli spaces of connections, isomonodromic deformations and Painlev\'e equations"

Abstract:I will summarize my joint works on geometry of moduli spaces of connections on curves with regular and irregular singularities. I will explain the classical Painlev\'e equations can be obtained from isomonodromic deformation of connections on curves. Then I will explain a few delicate points for proving the Painlev\'e property of differential equations arising from isomonodromic deformations of connections.

11:15-12:15 Shimpei Kobayashi (Hirosaki University), "Harmonic trinoids in complex projective spaces"

Abstract: I will talk about a construction of harmonic maps from a thrice-punctured sphere to complex projective space. Since such harmonic maps are characterized by the Toda field equations, integrable systems methods can be applied, that is, the problem of nonlinear PDE can be converted to a problem of linear ODE with parameter. Then the main difficulty is the monodromy problem, the global behavior of solutions of the linear ODE. I use generalized hypergeometric equations and solve the monodromy problem for particular cases.

LUNCH BREAK

13:45-14:45 Shinobu Hosono (Tokyo University), "Differential rings over the moduli spaces of Calabi-Yau manifolds"

Abstract: I will discuss some basic properties of certain differential rings on the moduli spaces of Calabi-Yau manifolds. Based on the rings, I will introduce/discuss the so-called BCOV (Bershadski-Cecotti-Ooguri-Vafa) holomorphic anomaly equations and also their solutions.

15:00-16:00 Martin Guest (Tokyo Metropolitan University), "The tt*-Toda equations (continued)"

Abstract: I will report on some progress in describing solutions of the tt*-Toda equations. These equations were introduced in the paper arXiv:1010.1889 (joint work with Chang-Shou Lin) as a generalization of certain equations considered by Cecotti and Vafa in the 1990s. They amount to a coupled system of Painleve III equations.

TEA/COFFEE BREAK

16:30-17:30 Claus Hertling (University of Mannheim), "Painleve III, radial sinh-Gordon and their isomonodromic connections as integrable twistor structures with additional data"

Abstract: I will discuss the solutions of the radial sinh-Gordon equation with the sign for cmc surfaces in Minkowski space. The solutions for this sign have singularities. The global geometry of these singularities is beautiful. The background for this are TERP-structures ( ~ integrable twistor structures with real structure, ~  noncommutative Hodge structures without Q-structure and without compatibility of Stokes structure and real structure), there results of Sabbah, T. Mochizuki and myself, and old results of Its+Novokshenov and McCoy+Tracy+Wu.