Type: Seminar

Distribution: World

Expiry: 6 Aug 2014

Auth: sarahl@p630x.pc (assumed)

A/Prof Anton Dzhamay from the University of Northern Colorado will give a talk on Wednesday 6th August at 1pm in Bosch Lecture Theatre 2. All welcome! TITLE: Discrete Schlesinger Transformations and Difference Painlevé Equations ABSTRACT: The main goal of the talk is to explain and compare the isomonodromic approaches to the study of Painlevé equations in the differential and difference cases. In the first part of the talk I will describe a discrete version of isomonodromic deformations of Fuchsian systems, called Schlesinger transformations. Similarly to the continuous case, such transformations give a large number of commuting flows on the moduli space of Fuchsian system (of a given spectral type), but in the discrete case we can write explicit equations governing the dynamic; we call these equations discrete Schlesinger equations. These equations can, again in exact parallel with the continuous case, be written in a Hamiltonian form on a slightly larger decomposition space (or the eigenvector space) of Fuchsian systems. When our phase space is two-dimensional, discrete Schlesinger transformations reduce to difference analogues of Painlevé equations. In the second part of the talk I will consider one example of such reduction that gives a difference Painlevé equation d-P(A_{2}^{(1)*}). I will emphasize the role played by Sakai’s geometric theory of discrete Painlevé equations in understanding the structure of this reduction, and in comparing discrete Schlesinger dynamic with other examples of difference Painlevé dynamic of the same type.