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.

In a very large monograph of he 70s Almgren provided a deep analysis of the singular set of area minimizing surfaces in codimension higher than 1. I will explain how a more modern approach reduces the proof to a manageable size and allows to go beyond his groundbreaking theorem.
 

A Willmore surface in the 3-dimensional Euclidean space is a critical point of the
square norm of the mean curvature of the surface.
The round spheres, the Clifford torus and the minimal surfaces are Willmore. For a
graph to satisfy the Willmore surface equation, its defining function is governed by
a fourth order non-linear elliptic equation. A classical theorem of Bernstein says
that an entire minimal graph must be a plane. We ask what happens to the entire
Willmore graphs. In this talk, I will discuss joint work with Tobias Lamm on the
finite energy case and with Yuxiang Li on the radially symmetric case.

In the last several years, there has been significant theoretical progress on understanding the average rank of all elliptic curves over Q, ordered by height, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as discuss generalizations in many directions (e.g., to other families of elliptic curves, higher genus curves, and higher-dimensional varieties) and some corollaries of these types of theorems. We will also describe recently collected data on ranks and Selmer groups of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt).