Ergodic theory is an active, central area of Modern Analysis. Its origins lie in the theoretical formulation for classical statistical mechanics at the end of the nineteenth century. The modern point of view derives from the profound mathematical theory of recurrence and ergodicity, as developed by Poincare, Birkhoff, von Neumann and other giants earlier in this century. As the subject has evolved and sophistication continually increased, ergodic theory has acquired close relationships with other branches of mathematics, such as dynamical systems, probability theory, functional analysis, number theory, differential topology, and differential geometry, and with applications as far ranging as mathematical physics, information theory, and computer design. The principal objects of study in ergodic theory are transformations of an underlying set or space. If for example, the transformation results from a differential equation, is time dependent, or more generally evolves according to a more extensive process than time, one speaks of a flow on the underlying geometry. This is the context of the modern theory of dynamical systems, in which a fundamentalcritical question is the classification of flows on homogenous spaces. Professor Ratner is one of the world leaders in the study of certain geometric flows and their dynamics. Her work on "rigidity" -- the classification of flows in terms of minimal data -- of horocycle flows is one of the top recent achievements in ergodic theory and dynamical systems. It comprises the most significant work to date in the classification of flows on homogeneous manifolds. Her work has produced examples and counterexamples, and it has brought an important geometrical point of view to the subject. In the current proposal she plans to study more general, so-called unipotent, flows. She hopes to classify all Borel probability measures preserved by unipotent tranformations on finite volume homogeneous spaces; to investigate the rigidity properties of such transformations; and to establish exponential mixing for ergodic flows on unit tangent bundles of compact surfaces of variable negative curvature.
