The classical crystallographic groups of ordinary Euclidean space are basic examples of discrete subgroups of Lie groups. The crystallographic groups arising in the very rigid geometric structures known as locally symmetric spaces of higher rank have been classified by Margulis. Witte will investigate whether these important groups can be realized as symmetries of one-dimensional spaces. Witte will also investigate the crystallographic groups that arise in homogeneous spaces other than the usual symmetric spaces. 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.
