A smooth metric space is a Riemanian manifold together with a weighted volume. It is naturally associated with a weighted Laplacian. In this talk, I will discuss some recent results about function theoretic and spectral propeties of the weighted Laplacian and volume estimates for the volume and weighted volume. The results can be applied to study the shrinking gradient Ricci solitons and self-shrinker for mean curvature flows.

In Taubes' proof of the Weinstein conjecture, a main ingredient is the estimate on the spectral flow of a family of Dirac operators, which he used to obtain the energy bound. When the perturbation is a contact form, much evidence suggests that the asymptotic behavior of the spectral flow function is nicer. In this talk, we will explain how to improve the spectral flow estimate for some classes of contact forms.

In this talk, the relationship between integrable systems and invariant curve flows is studied. It is shown that many integrable systems including the well-known integrable equations and Camassa-Holm type equations arise from the non-stretching invariant curve flows in Klein geometries. The geometrical formulations to some properties of integrable systems are also given.

I will introduce a parabolic flow of almost K\"ahler structures,
providing an approach to constructing canonical geometric structures on symplectic manifolds. I will exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing my previous work on parabolic flows of Hermitian metrics. I will exhibit a long time existence obstruction for solutions to this flow by showing certain smoothing estimates for the curvature and torsion. Finally I will discuss the limiting objects as well as some open problems related to the symplectic
curvature flow.