Random matrix models and their applications

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This research project is directed at basic problems of the theory of random matrix models and their applications to statistical physics. The project concerns with different conjectures of universality of the scaling limits of eigenvalue correlation functions and with exactly solvable models of statistical physics and hydrodynamics. This includes: (a) the development of the Riemann-Hilbert approach to random normal matrix models with applications to Hele-Shaw flows in hydrodynamics and Laplacian growth; (b) the development of the Riemann-Hilbert approach to random matrix models with external source; (c) the exact solution of the six-vertex model with domain wall boundary conditions in ferroelectric and antiferroelectric phases; (d) the study of the phase separation in the six-vertex model, and others.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 develops powerful mathematical methods to solve the problems of scaling and universality in the theory of random matrices and their applications. It involves methods of different areas of mathematics: analysis, theory of integrable systems, probability theory, semiclassical asymptotics, complex analysis, etc. The research project 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 the theory of knots and links and related problems in molecular biology, growth models, statistical data analysis, and others.