Alexander Its (Indianapolis)

A conformal block approach to the connection problem of the third Painleve equation

Abstract: In the recent series of papers of O. Gamayun, N. Igorov, O. Lisovyy, A. Shchechkin, J. Teschner, and Yu. Tykhyy a novel approach to the analysis of the Painleve equations has been suggested. The method is based on the Virasoro conformal block representation of the relevant tau-functions, and it allows, in particular, to evaluate the constant term in the asymptotics of the Painleve VI tau-function. It should be mentioned that the evaluation of the constant terms in the asymptotics of tau-fucntions is a very serious challenge for the usual Riemann-Hilbert isomonodromy method. In the talk, the conformal block technique will be extended to the case of the third Painleve equation. A new feature which needs to be taken care of is the presence of the irregular singularities in the problem. This is the joint work with O. Lisovyy and Yu. Tykhyy.