I will discuss natural energy functionals related to the
existence of holomorphic structures on vector bundles and show how
inauspicious Hodge data implies blow up of minimizing sequences.
Grassmann embeddings and an analytic perspective on stability in the
sense of Gieseker and Mumford plays an important role.

The classical Kauffman bracket is an invariant of knots in
space. It can be generalized to knots drawn on a surface. I will
discuss surprising properties of these generalized Kauffman
brackets.

This is a joint work with Tian. We study the structure of the limit space of a sequence of almost
Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the
initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the
$L^1$-sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a
sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein
manifolds. As applications, we can apply our structure results to study the
properties of K\"ahler manifolds.

A Hodge class on a smooth complex projective variety gives rise to an associated hermitian line bundle on a Zariski open subset of a complex projective space P^n.  I will discuss recent work with P. Brosnan which shows that the Hodge conjecture is equivalent to the existence of a particular kind of degenerate behavior of this metric near the boundary.

We report on recent and ongoing work with Zhou Gang and I.M.
Sigal in which we prove that all MCF neckpinches are asymptotically
rotationally symmetric. Combined with recent work of other authors, this
represents strong evidence in favor of the conjecture that MCF solutions
originating from generic initial data are constrained to one of exactly
two asymptotic singularity profiles.