Painleve's equation, tau functions and linear systems

Abstract: 

Tau functions can be introduced as Fredholm determinants of Hankel integral operators, or products of Hankel operators. Such tau functions generalize the classical notion of a theta function, as in complex tori. In this talk, we show how to define tau functions directly from linear systems in a manner that reveals the algebraic properties of the tau functions. As an illustration, we consider tau functions related to the Painleve equation PI, which is used to generate solutions of KdV.