Grant: Discovery Grants Program - Individual

The project is devoted to the study of the spectral theory of flat surfaces (i. e.flat two-dimensional Riemannian manifolds with conical, cylindrical or Euclidean ends). Flat surfaces provide a source of nontrivial examples, where the general methods of the analysis on manifolds with singularities and the scattering theory can be tested and improved. Among the main objectives of our project is the investigation of the relation of various spectral characteristics of the scalar Laplacians in singular flat metrics (special values of the zeta-function, individual eigenvalues, determinant, scattering matrix, etc) to the moduli of the underlying Riemann surface or to a choice of the corresponding flat metric. We are also planning to study the extremal problems for these spectral characteristics (mostly the determinant of the Laplacian) considered as functionals on the moduli spaces. Potentially, the results of our project can be used to get important information about the geometric structure of various moduli spaces, in particular, the Hurwitz space (the moduli space of meromorphic functions over Riemann surfaces) and the moduli space of meromorphic differentials over Riemann surfaces. .