Research Statement Aaron Brown 1

of a manifold M . One then studies properties of f under iteration. Some general aims of the field are to understand the asymptotic and statistical properties of the orbits of f , identify and classify invariant geometric structures, and understand changes in dynamical behavior under small perturbations. Iteration of a diffeomorphism f : M → M generates an action of the 1-parameter group Z. Given an infinite, finitely-generated discrete group Γ, we may similarly consider actions of Γ by diffeomorphisms on a manifold M ; that is, we may consider homomorphisms α : Γ→ Diff r (M ) from Γ to the group of C r diffeomorphisms of M . Typically one only considers actions α : Γ→ Diff r (M ) with infinite image α(Γ). My recent research focuses primarily on actions of large discrete groups Γ. A common approach in my research is to study actions of large groups by restricting to a 1-parameter subgroup of Γ and using techniques and tools developed to study classical 1-parameter dynamical systems. For instance, two key tools used in my work on Zimmer’s conjecture are Lyapunov exponents and metric entropy—objects from classical dynamical systems used to measure asymptotic exponential divergence and exponential complexity of a diffeomorphism. I am particularly interested in rigidity phenomena for group actions; many questions in this research program are summarized by the following problems. In full generality, such problems are intractable. My research focuses on such problems under additional dynamical hypotheses on the action or geometric hypotheses on the acting group.