My recent work involves the use of careful analysis to study the asymptotic behavior of a number of 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 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 a semiclassical limit. Due to the analytic dependence on an eigenvalue parameter, complex analysis plays an important role in the asymptotics.
Feel free to visit my webpage (see link above) for more information on the following topics of research:
Singular asymptotics for nonlinear waves.
Integrable systems, methods of algebraic geometry, and Riemann-Hilbert problems.
Complex analysis, orthogonal polynomials, and problems of approximation theory.
Applications in planar waveguide optics, optical fiber telecommunications, and random matrix theory.
