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 : A local obstruction to finding an Einstein metric in a conformal class is the nonvanishing of the Bach tensor, which is dened as the gradient of the Weyl curvature functional S|W|^2. On a Kahler manifold there are no other obstructions, so any Bach-flat Kahler metric is locally conformally Einsteinian. In addition, the conformal factor is geometrically interesting and sometimes controllable. This talk will describe the results of a 2008 paper with X. Chen and C. LeBrun, where circumstances under which a Kahler manifold is Bach-flat were established, and where it was shown that these conditions hold for a certain Kahler metric on CP2#-2CP2 with non-zero conformal factor, establishing for the first time an Einstein metric on CP2#-2CP2.

Villa : In this talk we discuss how the motivic zeta functions introduced by Denef and Loeser generalize and help to study Hodge theoretic invariants associated to singular points of hypersurfaces. In particular, we focus in the case of irreducible hypersurface quasiordinary singularities.

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