Cocycles and Cohomological Equations Play a Key Role in Ergodic Theory and Dynamics

Cocycles and cohomological equations play a key role in ergodic theory and dynamics. It is an active area of research to study what structure the set of cohomology classes, with respect to a group action, might have. For the so-called "higher rank" actions by commuting partially hyperbolic automorphisms, the cohomology classes are rigid, as shown by A. Katok and R. Spatzier. We will consider a few specific examples of smooth actions which are more aptly described as parabolic using techniques from representation theory.

In the first talk, we will give an introduction to the representation theory of semisimple Lie groups. We will discuss basic definitions, the theory for compact groups, and then turn to the two simplest non-compact semisimple groups, SL(2,R) and SL(2,C).

In the second talk, we will discuss the ability of representation theory to study cocycles over the horocycle and geodesic flows on Riemann surfaces. We characterize the obstructions to the solution of the cohomological equation for smooth cocycles.

In the last talk, we will show for several examples, that the first cohomology group for higher rank abelian actions by parabolic operators is isomorphic to the first cohomology group for the larger ambient action, and, therefore, in many cases vanishes.