Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré
and Geometrization Conjectures, some of its basic properties are still unknown. For
example it has been an open question whether the surgeries eventually stop to occur
(i.e. whether there are finitely many surgeries) and whether the full geometric
decomposition of the underlying manifold is exhibited by the flow as times goes to infinity.

In this talk I will show that the number of surgeries is indeed finite and that the
curvature is globally bounded by C t^{-1} for large t. Using this curvature
bound it is possible to give a more precise picture of the long-time behavior of the
flow.

It has been a challenging problem to studying the existence of Kahler-Einstein metrics on Fano manifolds. A Fano manifold is a compact Kahler manifold with positive first Chern class. There are obstructions to the existence of Kahler-Einstein metrics on Fano manifolds. In these lectures, I will report on recent progresses on the study of Kahler-Einstein metrics on Fano manifolds. The first lecture will be a general one. I will discuss approaches to studying the existence problem. I will discuss the difficulties and tools in these approaches and results we have for studying them. In the second lecture, I will discuss the partial C^0-estimate which plays a crucial role in recent progresses on the existence of Kahler-Einstein metrics. I will show main technical aspects of proving such an estimate.

In large dimensions, the only known compact, simply connected Riemannian manifolds with positive sectional curvature are spheres and projective spaces. The natural metrics on these spaces have large isometry groups, so it is natural to consider highly symmetric metrics when searching for new examples. On the other hand, there are many topological obstructions to a manifold admitting a positively curved metric with large symmetry. I will discuss a new obstruction in this setting. This is joint work with Manuel Amann (KIT).

Conference website
http://www.math.uci.edu/~scgas/scgas-2014/2014.php

The L2 norm of the Riemannian curvature tensor is a natural energy to associate to a Riemannian manifold, especially in dimension 4.  A natural path for understanding the structure of this functional and its minimizers is via its gradient flow, the "L2 flow."  This is a quasi-linear fourth order parabolic equation for a Riemannian metric, which one might hope shares behavior in common with the Yang-Mills flow.  We verify this idea by exhibiting structural results for finite time singularities of this flow resembling results on Yang-Mills flow.  We also exhibit a new short-time existence statement for the flow exhibiting a lower bound for the existence time purely in terms of a measure of the volume growth of the initial data.  As corollaries we establish new compactness and diffeomorphism finiteness theorems for four-manifolds generalizing known results to ones with have effectively minimal hypotheses/dependencies.  These results all rely on a new technique for controlling the growth of distances along a geometric flow, which is especially well-suited to the L2 flow.