My initial interests were in dynamical systems, with applications to the statistical mechanics of disordered systems, especially the study of phase transitions. Since then I have worked in several areas, following problems where they lead.
I am a theorist: I develop models along with algorithms and rigorous justification. Recent interests include:
Phase transitions and non-equilibrium statistical mechanics.
Random matrix theory and the probabilistic analysis of numerical algorithms.
Statistical theories of turbulence.
Embedding theorems and their applications in geometry, learning and physics .
My primary current goal is the development of a probabilistic approach to the embedding theorems of Nash and Whitney. My fascination with these theorems is rooted in the conviction that embedding is best seen as a form of information transfer between a noise source and an observer and that the only fundamental limits on information transmission are those provided by Shannon's channel coding theorem. This idea is unexpected if one takes the historical view of these theorems, but it is stimulated by several recent developments in mathematics. These include analogies between embedding and turbulence, the challenge of constructing random geometries for several applications in the sciences, and the use of the embedding theorems in learning theory.
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 five years, I have developed several new classes that bridge traditional areas of applied mathematics (PDE, dynamical systems) with models and insights from information theory and learning theory.Click on the link above for lecture notes.
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.
The primary purpose of academic publication is the communication of knowledge not the protection of prestige. I only review or submit papers to low-cost journals.
I am also a contributing writer for The Mathematical Intelligencer .
I am always happy to talk with undergraduate students at Brown. The best day to meet with me is on Friday.
If you're looking for an undergraduate research project with me, here are my parameters. I have advised several honors theses and I'd be happy to discuss this with you. 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 have no other research opportunities for undergraduates. In particular, I do not work with undergraduate students during the summer break, nor do I work with any students who are not enrolled at Brown University.
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.
Image courtesy Jessica Wynne .