Georgia Institute of Technology College of Sciences

The study of actions of subgroups of SL(k,\R) on the space of unimodular lattices in \R^k has received considerable attention since at least the 1970s. The dynamical properties of these systems often have important consequences, such as for equidistribution results in number theory. In particular, in 1984, Margulis proved the Oppenheim conjecture on values of indefinite, irrational quadratic forms by studying one dimensional orbits of unipotent flows. A more complicated problem has been the study of the action by left multiplication by positive diagonal matrices, A. We will discuss the main ideas in the work of Einsiedler, Katok and Lindenstrauss where a measure classification is obtained, assuming that there is a one parameter subgroup of A which acts with positive entropy. The first talk is devoted to completing our understanding of the unipotent actions in SL(2,\Z)\ SL(2,\R), a la Ratner, because it is essential to understanding the "low entropy method" of Lindenstrauss. We will then introduce the necessary tools and assumptions, and next week we will complete the classification by application of two complementary methods.