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 analysis of numerical algorithms.
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. While self-assembly today is principally driven by experimental advances, it is a promising field for mathematical contributions. Here is a link to a brief documentary explaining part of our work.
I currently serve on the editorial boards of the following journals.
Applied Mathematics Letters
Journal of Nonlinear Science
Physica D: Nonlinear Phenomena
SIAM Journal on Mathematical Analysis
As everyone who is involved in the publishing of mathematics knows, the cost of journals has skyrocketed, while most of the actual work involved in publishing (i.e. writing, editing and even typesetting papers) has been pushed onto authors. I find the situation as offensive as most, but I don't support boycotts of Elsevier or Springer. The main reason is that both these publishers support journals such as Physica D and the Journal of Nonlinear Science that are dedicated to interdisciplinary research in a way that the low-cost journals published by the AMS and SIAM are not. If you feel that your work is in this vein, and you would rather not support Physica D or JNLS, I'd be happy to consider your paper for QAM (which is distributed by the AMS).
I am also a contributing writer for The Mathematical Intelligencer .
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.
If you are a prospective student, and you'd like to work with me, please note that students at Brown are admitted to the graduate program, and are free to choose their advisor, so my "queue" should play no role in your decision to apply to Brown. Student plans often change (as they should) after some time in graduate school.