Complex Analysis, Dynamics and Geometry Seminar

The theory of Diophantine approximation is underpinned by Dirichlet's fundamental theorem. Broadly speaking, the main questions in the theory concern quantifying the prevalence of points with exceptional behavior with respect to Dirichlet's result. Badly approximable, very well approximable and Dirichlet-improvable points are among the most well-studied such exceptional sets. The work of Dani and Kleinbock-Margulis places these questions within the broader realm of homogeneous dynamics. After a brief overview of the subject and the motivating questions, I will discuss new results giving sharp upper bounds on the Hausdorff dimension of divergent orbits under certain partially hyperbolic homogeneous flows emanating from proper submanifolds and fractals on leaves of the unstable foliation. Applications towards Diophantine approximation will be presented. Speaker(s): Osama Khalil (OSU)