Riemann-Hilbert approach to the cylindrical Toda equations (I)
11:00:00 - 12:00:00
308 , Mathematics Research Center Building (ori. New Math. Bldg.)
The cylindrical Toda equations are a system of nonlinear o.d.e. which generalize the third Painleve (PIII) equation. Like the PIII equation, they have an isomonodromic deformation interpretation, and their solutions can be studied by the Riemann-Hilbert method. Our main interest is the "smooth solutions", as these are particularly important in geometry (globally defined harmonic maps or harmonic bundles) and physics (tt* equations for deformation of 2D SUSY quantum field theories). It turns out that the monodromy data (Stokes and connection matrices) are computable, due to the highly symmetric nature of the equations, and the Riemann-Hilbert problem can be solved. We shall explain this in the first lecture. In the second lecture we compare with other approaches such as elliptic p.d.e. theory and the Fredholm determinant method of Tracy-Widom. This is joint work with Alexander Its (IUPUI) and Chang-Shou Lin (NTU).