Mean field equation, Isomonodromic deformation and Painleve VI equation (I)


Ting-Jung Kuo

15:40:00 - 16:40:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE. In talk (I), we will mainly talk about the isomonodromy theory on the moduli space of elliptic curves. Firstly, we will introduce so called the generalized Lame equation defined on a torus which is motivated from the mean field equation. By studying the isomonodromy deformation of the generalized Lame equation, we discover that the elliptic form of Painleve VI equation is equivalent to a Hamiltonian system which is the isomonodromic equation defined on the moduli space. As an application, we show that the generalized Lame equation will converge to a classical one. In talk (II), we will apply this isomonodromy theory to establish the surprising connection surprising connection of real solutions to Painlevé VI with bubbling solutions of the mean field equations. Here a solution is called real if its associated monodromy representation is unitary. This connection provides us a geometric interpretation of solutions to Painlevé VI, and enables us to employ the PDE result to prove that, among other things, any real solution λ(t) to some Painlevé VI and its Backlünd transformation λ⁻¹(t), (1-λ(t))⁻¹ and (t-λ(t))⁻¹ are smooth for t∈R\{0,1}. This is a joint work with Prof. Chang-Shou Lin.

For material related to this talk, click here.