Ever since the 1970's, holomorphic twistor spaces have been used to study the geometry and analysis of their base manifolds. In this talk, we will introduce integrable complex structures on twistor spaces fibered over complex manifolds that are equipped with certain geometrical data. The importance of these spaces will be shown to lie, for instance, in their applications to bihermitian geometry, also known as generalized Kahler geometry. (This is part of the generalized geometry program initiated by Nigel Hitchin.) By analyzing their twistor spaces, we will develop a new approach to studying bihermitian manifolds. In fact, we will demonstrate that the twistor space of a bihermitian manifold is equipped with two complex structures and natural holomorphic sections as well. This will allow us to construct tools from the twistor space that will lead, in particular, to new insights into the real and holomorphic Poisson structures on the manifold. 

In manifolds with special holonomy, it is interesting to
study calibrated submanifolds, which are volume minimizer in their
homology classes. We study the calibrated submanifolds and mean
curvature flow in several famous local models of manifolds with
special holonomy. These model spaces are all total spaces of some
vector bundles, and the zero section is a calibrated submanifold. We
show that the zero section is the only compact minimal submanifold,
and is dynamically stable under the mean curvature flow. This is a
joint work with Mu-Tao Wang.

[Cancellation due to weather-related flight delays.  Talk to be rescheduled.]

The asymptotic expansion for the Bergman kernel has
important applications in complex analysis. Short-time asymptotic
expansion of the heat kernel played an important role in spectral
geometry. We will present our work on Feynman diagram formulas for
the coefficients in the asymptotic expansion of Bergman and heat
kernels on Kahler manifolds and their applications.

The Strominger system is a system of PDEs derived by Strominger in his
study of compactification of heterotic strings with torsion. It can be
thought of as a generalization of Ricci-flat metrics on non-Kähler
Calabi-Yau 3-folds. We present some new solutions to the Strominger
system on a class of noncompact 3-folds constructed by twistor
technique. These manifolds include the resolved conifold
Tot(O(-1,-1)->P1) as a special case.

The existence of the Ricci flow on manifolds with unbounded curvature remains an open problem. I'll talk about recent progress on this problem where the manifolds satisfy appropriate additional assumptions, and I'll show a few immediate applications.