Mathematical Sciences: Ergodic Theory of Unipotent          Translations on Homogeneous Spaces
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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.
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