Schedule of talks:

Abstracts

Paul : A long standing open problem in complex geometry is to find necessary and sufficient conditions for the existence of Kahler Einstein metrics with positive scalar curvature. Conjecturally the existence of such a metric is connected to the projective algebraic geometry of the manifold in question. On the other hand, necessary and sufficient conditions for existence can be formulated in terms of K-energy bounds (this is due to Tian) on the space of Kahler potentials. The question is how these bounds can be deduced from algebraic geometry. Very recently, I have been able to show that the K-energy is bounded below along all "degenerations" if and only if the hyperdiscriminant polytope dominates the Chow polytope. I will discuss these matters in the talk.

Cappell :

Naber : The work of Gromov and Fukaya tells us that a Riemannian manifold with bounded geometry has bounded topology. More specifically if a complete manifold has curvature bounded by one then there is a dimensional constant r(n) so that every ball of radius r has known topology. If one works a little more it can also be said that the ball is Gromov Hausdorff close to a torus quotient of a possibly lower dimenional euclidean ball. The main purpose of this talk is to show that in the case of lower or bounded Ricci that a form of sharp converse holds to this statement. That is a space with Ricci bounds that looks like a space of bounded geometry does in fact have bounded geometry.

Yang :

Wenger : Gromov's compactness theorem for metric spaces asserts that every uniformly compact sequence of metric spaces has a subsequence which converges in the Gromov-Hausdorff sense to a compact metric space. This theorem has been of great importance in Riemannian and metric geometry and also other fields. I will show in this talk that if one replaces the Hausdorff distance appearing in Gromov's theorem by the filling volume or flat distance then every sequence of oriented k-dimensional Riemannian manifolds with a uniform bound on diameter and volume has a subsequence which converges in this new distance to a countably k-rectifiable metric space. In general, such a sequence does not have a subsequence which converges with respect to the Gromov-Hausdorff distance. The new distance mentioned above was first introduced and studied by Christina Sormani and myself. In the talk, which will be self-contained, I will also explain the basic properties of this distance, its relationship with other distances, and illustrate it by examples.

Weber :

Villa :

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