Description:

Title: Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature

Abstract: There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare.  Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting.  Our main theorem is  that a closed negatively curved analytic Riemannian manifold with infinitely many totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold.