Algebro-geometric methods in integrable systems, random matrices and spectral geometry

点击翻译按钮获取中文摘要

This research proposal is mainly focused on the application of the theory of matrix Riemann-Hilbert problem, algebro-geometric methods and the theory of quantum groups to models of classical and quantum gravity, Frobenius manifolds, the theory of random matrices and spectral theory of Riemann surfaces. We plan to use the algebro-geometric methods developed earlier to solve boundary value problem corresponding to superposition of Kerr black hole with rotating semi-infinite matter disc, and to find a periodic analog of Kerr black hole determining thereby new physically important exact solutions of Einstein's equations.We plan to develop further the theory of quantum Einstein-Rosen waves with two polarization with the aim of making new advances towards a formulation of quantum gravity. We also plan to study physically relevant asymptotic expansions in hermitian matrix models, which could be used in numerous areas of applications of the random matrix theory from statistics to quantum gravity. Finally, we hope to contribute to the spectral theory of Laplace operator on flat Riemann surfaces with trivial and finite holonomy groups.