Professor Semmes' research project will cover two areas. The first deals with harmonic analysis and geometric measure theory. It is concerned with certain quantitative notions of rectifiability and their relationship to boundedness of singular integral operators. The second area deals with generalized Riemann mappings in several complex variables. The goal is to find classes of mappings defined by conditions much weaker than the requirement of being biholomorphic, so that there is a reasonable existence theory, but in such a way that these mappings reflect a large part of the complex structure. Professor Semmes' research in harmonic analysis involves the study of certain kinds of functions on infinite dimensional spaces that arise, for example, in the study of the boundary value problems of mathematical physics. His study of generalized Riemann mapping functions is an attempt to better understand the geometry of functions of many complex variables, a subject infinitely more complicated than the theory of one complex variable.
