Broadly: analysis and applied mathematics. My current work is on the formation and propagation of disorder in physical models, including:
Kinetics of phase transitions and models of domain coarsening.
Integrable systems and random matrix theory.
Statistical theories of turbulence.
One of my main goals is to construct ensembles of solutions to partial differential equations whose fluctuations are in accordance with empirical observations (for example, to describe the crumpling of surfaces or turbulence in fluids). Another theme in my work is the use of integrable systems. While integrable systems have traditionally been tied to very special nonlinear wave equations such as KdV and NLS, it is now clear that they link random matrices, growth models, quantum gravity and even number theory, in tantalizing and mysterious ways.
I am also fascinated with the idea that biology can inspire nanotechnology. I collaborate with labs that use self-assembly to design containers, devices and supramolecular assemblies. Here is a link to a brief documentary explaining part of our work.
I am happy to serve as an advisor for an honors thesis. My typical expectation is that you work with me for at least two semesters, commiting time equivalent to at least one class per semester. I do not fund summer "internships" for international students.