This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces. Professor Ratner is a world leader in ergodic theory. She will continue her work on the "rigidity" of unipotent subgroups of connected Lie groups. In particular, she will attempt to prove a conjecture of Raghunathan which states that every such subgroup is "algebraic". Such a result would have important implications for the diverse fields of ergodic theory, geometry, and number theory.
