I have broad interests in applied mathematics. Most of my work is on the dynamics of disordered systems, especially:
Phase transitions (especially in materials science) and non-equilibrium statistical mechanics.
Integrable systems and random matrix theory.
Statistical analysis of numerical algorithms.
Models of turbulence.
These areas are linked through a common viewpoint. The kinetics of phase transitions and models of turbulence are important examples of dynamical systems in high dimensions. Surprisingly, several numerical algorithms may also be viewed as dynamical systems acting in high dimensional spaces and behave like glassy systems. This approach suggests new questions and (at the technical level) new equations and estimates that I hope will be valuable to practitioners in different fields.
My ideals are those of a mathematical physicist. I am constantly amazed by the mathematical beauty of fundamental physical models. Dirac's dictum that all physical principles must have mathematical beauty, seamlessly coexists with Cantor's view that the essence of mathematics resides in its freedom. We have no understanding of why this is the case, but it is a gift of knowledge that belongs to us all.
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.
I teach classes undergraduate and graduate classes in many areas of applied mathematics. In the past few years, I have developed several new classes.
I currently serve on the editorial boards of the following journals.
Journal of Nonlinear Science (end 12/2016).
Physica D: Nonlinear Phenomena (end 12/2016).
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.