Abstract

The results presented in this thesis contribute to the understanding of the interplay between the geometry of Riemann surfaces and their moduli space and the theory of integrable systems. First, in Chapter 2, we present a new degeneration of Fay’s trisecant identity leading to algebro-geometric solutions of the Schwarzian Kadomtsev-Petviashvili equation in terms of Riemann theta functions. Then, in Chapter 3, we introduce a new set of log-canonical coordinates on the SL(2,C) character variety of compact Riemann surfaces; these coordinates are constructed by combining shear type coordinates with length-twist type coordinates.