An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: An Introduction

On the middle convolution for Fuschian systems

15th September 2006

Author: Galina Filipuk (Kumamoto University)

Abstract

In the talk I shall explain the algorithm of Dettweiler and Reiter who generalized the Katz middle convolution functor.

Middle convolution is an operation for Fuchsian systems of differential equations which preserves rigidity (and, hence, the number of accessory parameters) but changes the rank and monodromy group.

In the simplest case of the sixth Painleve equation which describes monodromy preserving deformations of the rank 2 Fuchsian system with four singularities on the projective line the algorithm is applied to derive the Okamoto birational transformation. Next I shall discuss the invariance of deformation equation under middle convolution, which is a join work with Yoshishige Haraoka.