We present a joint work with Richard Bamler.
We consider Ricci flows that satisfy certain scalar curvature bounds. It is found that the time derivative for the solution of the heat equation and the curvature tensor have better than expected bounds. Based on these, we derive a number results. They are: bounds on distance distortion at different times and Gaussian bounds for the heat kernel, backward pseudolocality, L^2-curvature bounds in
dimension 4.

In this talk I will describe a cohomological formula for a higher
index pairing between invariant elliptic differential operators and
differentiable group cohomology classes. This index theorem generalizes the
Connes-Moscovici L^2-index theorem and its variants. This is joint work with
Markus Pflaum and Hessel Posthuma.

We discuss certain inequalities for the Henneaux-Teitelboim total
energy-momentum for asymptotically anti-de Sitter initial data sets
which are asymptotic to arbitrary t-slice in anti-de Sitter spacetime. We
also give the relation between the determinant of the energy-momentum matrix
and the Casimir invariants. This is a joint work with Y. Wang and X. Zhang.

Closed quasi-Fuchsian subsurfaces of closed hyperbolic
3-manifolds constructed by J. Kahn and V. Markovic have played a crucial
role in the recent proof of the Virtual Haken Conjecture. In this talk, we
will investigate the techniques and construct homologically interesting
possibly bounded quasi-Fuchsian subsurfaces in closed hyperbolic
3-manifolds. We will focus on extending the geometric and topological
aspects from work of Kahn-Markovic, and will discuss further questions.
This is joint work with Vladimir Markovic.

Bott-Chern cohomology is a refinement of de Rham cohomology on
complex manifolds. We shall discuss the limit of Bott-Chern cohomology in
terms of hypercohomology for semistable degeneration of complex manifolds.
As an application, we show that nonkahler Calabi-Yau 3-folds obtained by
conifold transition satisfy d d\bar lemma, hence admit a Hodge decomposition.