Conferència SCM

Abstract:

The Painlevé differential equations play a universal role in many applications, occurring in classical areas such as fluid dynamics, plasma physics, optics, and many modern areas including statistical mechanics, random matrix theory, topological field theory, quantum gravity and quantum field theory. For this reason, their solutions are regarded as the modern counterpart of widely used traditional special functions, such as Bessel, Airy, hypergeometric and elliptic functions.

The mathematical theory of such nonlinear ODEs is extraordinarily rich and profound. To take just one direction, we mention their appearance as compatibility conditions for a pair of linear (differential or difference) matrix equations, referred to as a Lax pair. The Lax pair can be expressed as two matrix linear equations, one evolving in x, which does not appear at all in the corresponding nonlinear equation, and one evolving in t, which is the independent variable of the associated nonlinear equation. The information that characterises the x-linear problem is characterised by a set of data, called monodromy data, and the surprising discovery made a century ago is that it remains invariant under deformation by t. The invariant data form a monodromy manifold.

In this talk I will explore how to describe monodromy manifolds by Segre surfaces,  an intersection of two Quadric in 4-dimensionalProjective space. They are Rational surface isomorphic to a projective line blown up in 5 points with no 3 on a line, and generically have 16 rational lines.

Intriguingly, such lines on the Segre surface correspond to special solutions of the Painlevé differential equation with a-typical asymptotic behaviors.