Dmitry Korotkin

Laplace operator on flat Riemann surfaces and geometry of moduli spaces.

The determinant of Laplace operator on flat Riemann surfaces with conical singularities can be expressed in terms of a
special isomonodromic tau-function. This tau-function can be
expressed in terms of prime-forms and theta-functions on the
Riemann surface. Globally this tau-function is essentially a section of the Hodge line bundle. By computing the divisor of the tau-function, which turnes out to be supported only on the boundary of corresponding moduli spaces, we can express the Hodge class in terms of boundary divisors on moduli spaces
of holomorphic differentials on Riemann surfaces and Hurwitz spaces. This is a joint work with A.Kokotov and P.Zograf.