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.

In 1982 Calabi proposed studying gradient flow of the L^2 norm
of the scalar curvature (now called Calabi flow) as a tool for finding
canonical metrics within a given Kahler class. The main motivating
conjecture behind this flow (due to Calabi-Chen) asserts the smooth long
time existence of this flow with arbitrary initial data. By exploiting
aspects of the Mabuchi-Semmes-Donaldson metric on the space of Kahler
metrics I will construct a kind of weak solution to this flow, known as a
minimizing movement, which exists for all time.

We first survey some development regarding the study of holomorphic functions on manifolds. We insist mostly on Liouville theorems or, more generally, dimension estimates for the space of polynomially growing holomorphic functions. Then we present some recent joint work with Jiaping Wang on this topic. Our work is motivated by the study of Ricci solitons in the theory of Ricci flow. However, the most general results we have do not require any knowledge of curvature.