I am fascinated by the applicability of mathematical ideas to ``real world'' problems, and by how applied sciences in turn influence development of mathematics. In my research, I am interested in using techniques from applied analysis, dynamical systems, PDE, algebra and scientific computation to study interdisciplinary problems.
I have a multidisciplinary background. As an undergraduate, I majored in mathematics and chemistry. My math training then was more number theoretic in nature. For my senior thesis, I studied the distribution of quadratic residues. For my chemistry thesis, I studied radiative forcing by sulfate aerosols on global climate. In my graduate studies at the University of Chicago, I took classes from various departments to broaden my horizon. Among these classes are classical mechanics from Department of Physics; fluid mechanics, geophysical fluid dynamics, singular perturbation from Department of Geophysical Sciences; and stochastic processes from Department of Statistics. Apart from the standard math classes, I took dynamical system (both general and Hamiltonian), singular integrals, PDE, ergodic theory, finite element methods, applied analysis, and others.
For my dissertation I investigate the elliptical instability of a rotating, inviscid, incompressible fluid in an ellipsoid; normal forms for reversible dynamical systems; and analysis of algebraic structure of solutions to specific differential equations.
I would like to extend my dissertation project to examine the relevance of our results to elliptical instability in real fluids, where viscosity is always present. I am interested in instabilities of geophysical and atmospheric flows. Another potential application is to astrophysical flows and stellar structure, especially the fission theory of binary stars.
I am also interested in the dynamics, bifurcations and collective behavior in extended systems. In order to study them either analytically or numerically, we need to reduce the number of modes by separating the slow from the fast modes. Normal forms and center manifold reduction have been used to obtain amplitude equations or effective equations for slow modes e.g. the complex Ginzburg-Landau arises from the normal form for a Hopf bifurcation. I would like to apply these techniques in problems arising in atmospheric science and oceanography.
Another fascinating problem is the behavior of complex systems, which can spontaneously organize, and form coherent structures and patterns on multiple scales. This is a result of the system being governed by multiple physical processes, each operating on an independent scale. A wide variety of systems from physics, biology, fluids and material science exhibit such behavior. I would like learn about recent development of analytic and numerical methods for such problems and to be involved in modeling, analysis and computation.
The above is a sampling of problems I am aware of and am interested in. However, there must be a wide variety of interesting problems that I have not yet been exposed to. At this early stage of my research career, I would appreciate every opportunity to explore and learn about new problems, ideas and methods.