|XXVIII Workshop on Geometric Methods in Physics||28.06-04.07.2009|
Riemann--Hilbert problems: Schlesinger and Sato
Given any (sufficiently well-behaved) family of Remann--Hilbert problems where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form $\Omega$ on the deformation space (Malgrange's differential). Such a one--form has a pole where the deformation family meets the Malgrange Theta divisor, namely, the set of unsolvable RHP.
I will show how this one form, together with the notion of discrete Schlesinger transformation, leads naturally to a general form of Sato`s formula for the Baker-Akhiezer vector (even if no notion of Tau function is present). Time permitting I will give examples of application to the theory of Painlev\'e\ equations and T\"oplitz determinants.