In this talk, we discuss how to define the quasi-local conserved
quantities, the mass, angular momentum and center of mass, for a
finitely extended region in a spacetime satisfying the Einstein
equation. We start with the quasi-local mass and its properties and
then use the results to define other conserved quantities. As a
further application, we use the limit of the quasi-local conserved
quantities to define total conserved quantities of asymptotically flat
spacetimes at both the spatial and the null infinity and study the
variation of these quantities under the Einstein equation.

Free boundary minimal surfaces in the ball are proper branched minimal
immersions of a surface into the ball that meet the boundary of the ball
orthogonally. Such surfaces have been extensively studied, and they arise as
extremals of the area functional for relative cycles in the ball. They also
arise as extremals of an eigenvalue problem on surfaces with boundary. In
this talk I will describe uniqueness (joint work with R. Schoen) and
compactness (joint work with M. Li) theorems for such surfaces.

We will introduce two new Li-Yau estimates for the heat equation
on manifolds under some new curvature conditions. The first one is obtained
for n-dimensional manifolds with fixed Riemannian metric under the
condition that the Ricci curvature being L^p bounded for some p>n/2. The
second one is proved for manifolds evolving under the Ricci flow with
uniformly bounded scalar curvature. Moreover, we will also apply the first
Li-Yau estimate to generalize Colding-Naber's results on parabolic
approximations of local Busemann functions to weaker curvature condition
setting. This is a recent joint work with Richard Bamler and Qi S. Zhang.

In this talk, I will begin with introducing the method of
pseudo-holomorphic curves (which are defined by Cauchy-Riemnn type elliptic
systems) in the study of symplectic and contact topology. Then I will focus
on discussing the one studied by Yong-Geun Oh and myself recently, including
its potential, drawbacks and possible improvements towards the goal of a
better understanding in contact topology. (The similar elliptic system named
the generalized pseudo-holomorphic curves in symplectizations was introduced
by Hofer and studied by Abbas-Cieliebak-Hofer, Abbas in the proposal of
proving the Weinstein conjecture for dimension three.)

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal surfaces with bounded index on a given three-manifold might degenerate. We then discuss several applications, including some compactness results. Time permitting, we discuss how our strategy can be extended to ambient dimensions 4,5,6 and 7. (This is joint work with O. Chodosh and D. Ketover)