Given a  Kahler manifold, the smooth Calabi flow is the parabolic version of the constant scalar curvature equation. Given that this fourth order flow has a very undeveloped regularity and existence theory, J. Streets recast it as a weak gradient flow in the abstract completion of the space of Kahler metrics. In this talk we will show how a better understanding of the abstract completion gives updated information on the large time behavior of the weak Calabi flow, and how this fits into a well known conjectural picture of Donaldson. This is joint work with Robert Berman and Chinh Lu. 

We prove that the volume of a free boundary minimal surface
\Sigma^k \subset B^n, where B^n is a geodesic ball in Hyperbolic
space H^n, is bounded from below by the volume of a geodesic k-ball
with the same radius as B^n. More generally, we prove analogous
results for the case where the ambient space is conformally
Euclidean, spherically symmetric, and the conformal factor is
nondecreasing in the radial variable. These results follow work
of Brendle and Fraser-Schoen, who proved analogous results for
surfaces in the unit ball in R^n. This is joint work with Brian Freidin.

The Almgren-Pitts min-max theory is a Morse theoretical
type variational theory aiming at constructing unstable minimal
surfaces in a closed Riemannian manifold. In this talk, we will
survey recent progress along this direction. First, we will discuss
the understanding of the geometry of the classical Almgren-Pitts
min-max minimal surface with a focus on the Morse index problem.
Second, we will give an application of our results to quantitative
topology and metric geometry. Next, we will introduce the study of
the Morse indices for more general min-max minimal surfaces arising
from multi-parameter min-max constructions. Finally, we will
introduce a new min-max theory in the Gaussian probability space and
its application to the entropy conjecture in mean curvature flow.

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.