Mathematics Colloquia and Seminars

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In the last three decades or so, the Riemann-Hilbert approach has established itself as a powerful tool for obtaining precise asymptotic information about problems arising in various disciplines. These applications range from the theory of integrable nonlinear PDEs and random growth models, to random matrix theory, statistical physics and analytic number theory. Structured determinants are quite prevalent as they characterize principal objects of interest in various fields of research, especially in random matrix theory and statistical mechanics, and form bridges between these areas and the techniques of Riemann-Hilbert analysis.

In this talk, I will describe some recent developments in the understanding of several different types of structured determinants, both in their Riemann-Hilbert characterizations as well as their connection to problems in statistical mechanics, random matrix theory, and graph enumeration. More precisely, I will highlight

a) The role of Toeplitz, Toeplitz+Hankel, bordered Toeplitz and bordered Hankel determinants in statistical mechanics, spotlighting some recent developments in obtaining their large-size asymptotic behavior,

b) The role of Hankel determinants in random matrix theory, with a focus on the topological expansion and phase diagram of random matrices with complex quartic potentials. This gives rise to the solution of the associated combinatorial question: How many labeled four-valent graphs with j vertices can be embedded on a Riemann surface of genus g, and

c) Some novel aspects of the Riemann-Hilbert analysis, including the utilization of S-curves and the theory of quadratic differentials.