Integrable systems, moduli spaces and spectral geometry

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The goal of the proposed research is to apply methods originating in the theory of integrable systems (Riemann-Hilbert problems, tau-functions) to  moduli spaces, classical gravity and spectral geometry. In the area of moduli spaces we are going to address the following problems: analytical derivation of relations between various cohomology and homology classes (Witten cycles) on moduli spaces of Riemann surfaces; understanding of the "wall-crossing" behaviour of tau-functions on moduli spaces of quadratic and n-differentials on Riemann surfaces, and their transformation under the pentagram moves; application to Hodge integrals. This should help also to further elucidate the role of cluster algebras in the theory of moduli spaces. In the area of classical gravity the main goal is to find a periodic analog of Kerr solution; analytically such solution should corresponds to an infinite chain of solitons of Ernst equation.  In the area of spectral geometry we hope to develop a mathematically rigorous theory of determinants of Laplace operators in arbitrary flat metrics over Riemann surfaces with cylindrical outlets and study extremal problems for determinant of Laplacian in presence of conical singularities.