4867 East Hall
Phone: (734) 764-6419
- Mathematical Physics
My research is in mathematical physics, a branch of hard analysis. The physical systems I study mathematically usually arise either in elementary particle physics (and are quantum field theories) or in statistical mechanics. The most common mathematical tools used in the study are functional integration, and operator estimates in Hilbert space. In a sense, the flavor of the work is not dissimilar from parts of p.d.e. or numerical analysis: a complicated algorithm is developed, and one must prove its convergence - the proof of convergence is very complex and difficult, necessitating frequent readjustments in intimate details of the algorithm.
An expository article presenting some of my recent work is Quantum Field Theory in Ninety Minutes, Bull. A.M.S. 17, 93-103 (1987). I often suggest to individuals desiring to get the flavor of my work that they read: A New form of the Mayer Expansion in Classical Statistical Mechanics, with David Brydges, J. Math Phys. 19, 2064-2067 (1978). (If one doesn't like this paper, one will not like most of my research - it is a good exposure to the cluster expansion, my main analytic machine.)
Lately I have extended my interests to the study of the Navier-Stokes equations and turbulence, using a set of divergence-free vector wavelets I have constructed. Nice results on the Cauchy Problem with data in Morrey Spaces have already been obtained. More progress is expected here. Although most of my research is in hard analysis, there have been frequent tangential forays of quite a different flavor, for example, into algebraic topology, tree graph counting, and logarithmic Sobolev inequalities.