Professor Battle's research involves the use of wavelet analysis in quantum field theory. In this project, he will work on the infrared problem for spinor quantum electrodynamics in three space-time dimensions, and on the ultraviolet phase cell analysis of group-valued sigma models. Much of the enterprise of mathematical physics is concerned with analysis on spaces of functions, typically functions of several position and momentum variables. The functions represent possible states of whatever physical system is being modeled. Properties of the model are encoded by procedures for getting numbers out of these functions, or for transforming them into other functions. A general mathematical technique useful in studying these procedures is to expand in infinite series whose individual terms are in some sense basic and well-understood. Choice of the reference sequence of basic terms depends on context and objectives. In the models being studied here, for instance, what happens under change of scale is an important consideration. Professor Battle uses a strategy called wavelet analysis to produce expansions with good rescaling behavior. His project consists in applying this body of technique to difficult problems in mathematical physics.
