We will introduce two new Li-Yau estimates for the heat equation
on manifolds under some new curvature conditions. The first one is obtained
for n-dimensional manifolds with fixed Riemannian metric under the
condition that the Ricci curvature being L^p bounded for some p>n/2. The
second one is proved for manifolds evolving under the Ricci flow with
uniformly bounded scalar curvature. Moreover, we will also apply the first
Li-Yau estimate to generalize Colding-Naber's results on parabolic
approximations of local Busemann functions to weaker curvature condition
setting. This is a recent joint work with Richard Bamler and Qi S. Zhang.

In this talk, I will begin with introducing the method of
pseudo-holomorphic curves (which are defined by Cauchy-Riemnn type elliptic
systems) in the study of symplectic and contact topology. Then I will focus
on discussing the one studied by Yong-Geun Oh and myself recently, including
its potential, drawbacks and possible improvements towards the goal of a
better understanding in contact topology. (The similar elliptic system named
the generalized pseudo-holomorphic curves in symplectizations was introduced
by Hofer and studied by Abbas-Cieliebak-Hofer, Abbas in the proposal of
proving the Weinstein conjecture for dimension three.)

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal surfaces with bounded index on a given three-manifold might degenerate. We then discuss several applications, including some compactness results. Time permitting, we discuss how our strategy can be extended to ambient dimensions 4,5,6 and 7. (This is joint work with O. Chodosh and D. Ketover) 

Talk Schedule

2:00 - 2:50 PM     Karen Uhlenbeck (UT Austin)

3:10 - 4:00 PM     Gudlaugur Thorbergsson (Cologne)

4:00 - 4:30 PM     Reception

4:30 - 5:20 PM     Robert Bryant (Duke)

We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal conjectured bound in terms of the length of  the cut locus of a point on the surface. We also prove that the natural extension of the conjecture to general dimension holds among closed convex spherically symmetric Riemannian manifolds. Our results are based on a new symmetrisation procedure which we believe to be interesting in its own right.

This is joint work with Pedro Freitas accepted for publication in the Tohoku Mathematical Journal, preprint on http://arxiv.org/abs/1406.0811.