Margulis will continue his work on different application of the theory of subgroup actions on homogeneous spaces to number theory and Diophantine approximations. He will study discrete groups of affine transformations, and in particular the algebraic structure of cocompact discrete groups of affine transformations. He will also continue his work on the Zariski closure of the linear part of an arbitrary discrete group of affine transformations. 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.
