A one-parameter family of hypersurfaces in Euclidean space evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. It can be viewed as a geometric heat equation, i.e., it is locally moving in the direction of steepest descent for the volume element, deforming surfaces towards optimal ones (minimal surfaces). In this talk we will discuss some recent work on the local curvature estimate and convexity estimate for the star-shaped mean curvature flow and the consequences. In particular, star-shaped MCF is generic in the sense of Colding-Minicozzi. This is joint work with Robert Haslhofer. 

I will review some classical and new results about finite dimensional integrable
Hamiltonian systems, emphasizing the interplay between symplectic geometry
and spectral theory. The talk is aimed at a general audience.

The eigenvalues of the Laplacian encode fundamental
geometric information about a Riemannian metric. As an
example of their importance, I will discuss how they
arose in work of Cao, Hamilton and Illmanan, together
with joint work with Stuart Hall,  concerning stability
of Einstein manifolds and Ricci solitons. I will outline
progress on these problems for Einstein metrics with
large symmetry groups. We calculate bounds on the first
non-zero eigenvalue for certain Hermitian-Einstein four
manifolds. Similar ideas allow us estimate to the
spectral gap (the distance between the first and second
non-zero eigenvalues) for any toric Kaehler-Einstein manifold M in
terms of the polytope associated to M. I will finish by
discussing a numerical proof of the instability of the
Chen-LeBrun-Weber metric.

We show that for an immersed two-sided minimal surface in R^3,
there is a lower bound on the index depending on the genus and number of
ends. Using this, we show the nonexistence of an embedded minimal surface
in R^3 of index 2, as conjectured by Choe. Moreover, we show that the
index of an immersed two-sided minimal surface with embedded ends is
bounded from above and below by a linear function of the total curvature
of the surface. (This is joint work with Otis Chodosh)