Research Interests

As a mathematician, my interests lie in spectral theory, inverse spectral theory, and the theory of integrable systems. The modern era for integrable systems began with the solution of the Korteweg de Vries equation by Gardner, Greene, Kruskal and Miura in 1967. This led to the development of a variety of new mathematical techniques, and over time, and quite unexpectedly, these techniques have found applications in areas far beyond their dynamical origins. The applications include problems in algebraic geometry, numerical analysis, analytic number theory, combinatorics and random matrix theory. Amongst the techniques that have emerged, the Riemann-Hilbert method, which provides a non-commutative version of an integral representation for a problem at hand, has proved to be particularly useful. In 1993, together with Xin Zhou, I introduced a nonlinear version of the steepest descent method for Riemann-Hilbert problems that makes possible the asymptotic evaluation of Riemann-Hilbert problems as some parameter in the system, such as space or time, becomes large. As a consequence, a very broad variety of asymptotic problems in integrable systems and related areas, such as random matrix theory, have now been rigorously analyzed. Much of my research currently involves the application of the Riemann-Hilbert/steepest-descent method.

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Section 11: Mathematics