About
Recent research work involves the use of careful analysis to study the asymptotic behavior of a numberof different mathematical objects relevant in applied mathematics: solutions of certain partial differential equations that are canonical models for nonlinear wave dynamics, orthogonal polynomials of large degree and related special functions, and statistics of ensembles of random matrices. These different things may all be studied using techniques from the field of integrable systems. The fundamental concept in this field is that certain nonlinear equations may be viewed as compatibility conditions between two or more linear problems. In practice, using the integrability for asymptotic analysis means studying the associated linear problems, along with their related inverse problems, in various asymptotic limits. Due to the analytic dependence on a spectral parameter, complex analysis plays an important role in the asymptotics. Specific current research topics include the study of solutions of Painleve equations, universal phenomena in nonlinear wave propagation, and methods for the analysis of Riemann-Hilbert problems.
