This research project is directed on fundamental problems of the theory of random matrices and random polynomials and their applications, and on related problems in statistical physics. The cornerstone of the problems is different conjectures of universality, which state that as the size of a random matrix (or the degree of a random polynomial) approaches infinity, the correlations between properly scaled eigenvalues (or zeros) approach a universal limit. In the current project the PI continues his studies of the universality in random matrix models, random polynomials, and statistical physics. This includes: (i) The Riemann-Hilbert (RH) approach to double scaling limits in random matrix models. (ii) RH approach to random matrices with external source. (iii) Semiclassical asymptotics and RH approach to multi-matrix models. (iv) RH approach to the six-vertex model of statistical physics. (v) Scaling limits and universality in non-Gaussian ensembles of random polynomials and random algebraic varieties.
The project has an interdisciplinary character and it lies on the frontier between physics and mathematics. The problems of scaling and universality are central in many areas of modern science: theory of critical phenomena and phase transitions, statistical physics and quantum field theory, theory of quantum chaos, nonlinear dynamics, etc. This project is directed on development of powerful mathematical methods to the problems of scaling and universality in the theory of random matrices, random polynomials, and related topics. It involves different areas of mathematics: analysis, theory of integrable systems, probability theory, semiclassical asymptotics for systems of differential equations, complex analysis, etc. The research project under consideration has direct applications to various physical problems: combinatorial asymptotics related to quantum gravity, exactly solvable models of statistical physics, spin systems on random surfaces, theory of critical phenomena and phase transitions, quantum chaos. Possible further applications include theory of knots and links and related problems in molecular biology, growth models, statistical data analysis, and others.
