Grant: Discovery Grants Program - Individual

This research is focused on the spectral theory of Riemann surfaces, and theory of Hurwitz spaces and Frobenius manifolds. The main object of our research are compact polyhedral surfaces of an arbitrary genus (the boundary of a polyhedron with several holes may serve as an example). We are studying spectral properties of these surfaces and behavior of the spectrum under deformations. Another direction of our research are extremal problems related to polyhedral surfaces. As a toy example one can mention the following problem: to find among tetrahedrons of the fixed surface area the one for which the determinant of the Laplacian is maximal; the answer is the platonic tetrahedron and we are going to study further generalizations of this result. Another important objective of the research is studying of behavior of the Bergman tau-function near the boundary of various strata of the Hurwitz space. The Hurwitz space is the moduli space of meromorphic functions on compact Riemann surfaces, the Bergman tau-function is a naturally defined holomorphic function on this space. This object plays an important role in several domains of mathematics: theory of isomonodromic deformations, spectral theory, matrix models, theory of Frobenius manifolds. The successful fulfillment of this program will be a significant step ahead in the areas which are currently in the center of research activity in Canada and worldwide.