RESEARCH STATEMENT

My research falls into the areas of group theory, dynamical systems, and descriptive set theory. I study the geometric and combinatorial structure of groups and the structure and classification of their actions. Frequently there are strong relationships between these two, and I especially enjoy these connections. The most frequent topic I consider is that of a countable group G acting by measure-preserving transformations on a standard probability space (X,µ). Such an action G y (X,µ) is called probability-measure-preserving (p.m.p.), and the induced orbit equivalence relation is the equiva-lence relation EXG on X whose classes are given by the G-orbits. I am generally interested in the structure of p.m.p. actions and their classification up to isomorphism and up to orbit equivalence, i.e. the isomorphism type of the induced orbit equivalence relation EXG. These investigations fall within the field of ergodic theory. Frequently there are strong connections between groups and the structure of their p.m.p. actions. I enjoy many topics in group theory and enjoy studying countable groups themselves. Likewise, there are many connections between ergodic theory, topological dy-namics, and Borel dynamics, and I have interest in all three. The common thread which underlies my work on all of these topics is an approach to problems through combinatorial and geometri