An oriented hypersurface in a hyperkaehler 4-manifold naturally inherits a coclosed coframing.  Bryant showed that, in the real analytic case, any oriented 3-manifold with a coclosed coframing can always be locally “thickened” to a hyperkaehler 4-manifold, in an essentially unique way.  This raises the natural question: when can these 3-manifolds with this structure arise as the boundary of a hyperkaehler 4-manifold?  In particular, starting from a compact hyperkaehler 4-manifold with boundary, which deformations of the boundary structure can be extended to a hyperkaehler deformation of the interior?  I will discuss recent progress on this problem, which is joint work with Joel Fine and Michael Singer.

We consider the evolution by mean curvature flow of surface clusters,
where along triple edges three surfaces are allowed to meet under an equal angle
condition. We show that any such smooth flow, which is weakly close to the static
flow consisting of three half-planes meeting along the common boundary, is smoothly
close with estimates. Furthermore, we show how this can be used to prove a smooth
short-time existence result. This is joint work with B. White.

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.