Abstract : Localization is an important principle in mathematics and local index theorem is an example of such localization. We will discuss the application of the local index Theorem technique to the question of asymptotic expansion of Bergman kernel, which has played an essential role in Donaldson's work on extremal metrics. This is joint work with Kefeng Liu, Xiaonan Ma and Xiaowei Wang.

Abstract: Let $M^n$ be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that $M$ has positive spherical rank if along every geodesic there is a conjugate point at $t = \pi$ (geodesics are assumed to be parametrized by arc-length). In case $M$ has positive spherical rank, the equality discussion of the Rauch Comparison theorem implies that along every geodesic of $M$ we have a spherical Jacobi field i.e., a Jacobi field of the form $J(t) = \sin(t) E(t)$, where is $E(t)$ is a parallel vector field along the geodesic. This notion of rank is analogous to the notion of geometric rank for upper curvature bound 0 or upper curvature bound -1 studied by Ballmann, Burns, Spatzier, Eberlein, Hamenstaedt etc. In the case of spherical rank we show: Let $M^n$ be a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank. Then $M^n$ is isometric to a compact rank one symmetric space i.e., isometric to a sphere or projective space. This is joint work with Ralf Spatzier and Burkhard Wilking.

Abstract: Let $n$ be a dimension for which the classical Positive Mass Theorem holds in general relativity. Let $(X^{n+1}, g)$ be a conformally compact manifold whose Ricci curvature is bounded below by $-n$. We show that $(X^{n+1}, g)$ is isometric to the standard hyperbolic space provided the conformal infinity of $(X^{n+1}, g)$ is the standard sphere and $Ric(g) + n$ decays faster than $r^2$ near infinity, where $r$ is any boundary defining function. The proof is based on a quasi-local mass characterization of Euclidean balls, which is essentially a local version of the Positive Mass Theorem. This is a joint work with J. Qing.

Abstract. The Cheeger-Gromoll splitting theorem states that a complete Riemannian manifold with nonnegative Ricci curvature is isometric to the product of some Euclidean space and a manifold N where N has no lines, a line being defined as a smooth curve that minimizes distance in both directions. We will prove that any complete Riemannian manifold which does not have a property called the geodesic loops to infinity property has a line in its universal cover. We will then use this theorem along with the splitting theorem to discuss the relationship between the loops to infinity property and the structure of manifolds with nonnegative Ricci curvature. This is work of Christina Sormani.