Broadly speaking, there are two classes of inverse problems — those that are concerned with the analysis of PDEs, and those that are geometric in nature. In this talk, I will introduce the audience to these classes by highlighting two classical examples: Calderón’s problem for the PDE setting, and the boundary rigidity problem in the geometric setting. Then, I will present a proof overview for an inverse problem which uses techniques from both the PDE perspective and the geometric perspective. In particular, I will consider the following question:

Given any simple closed curve $\gamma$ on the boundary of a Riemannian 3-manifold $(M,g)$, suppose the area of the minimal surfaces bounded by $\gamma$ are known. From this data may we uniquely recover the metric $g$?

I will highlight several cases where the answer is yes. I will provide both a global and local uniqueness result given area data for a much smaller class of curves on the boundary. The key to showing uniqueness for the metric $g$ is that we can reformulate parts of the problem as a 2-dimensional inverse problem on an area-minimizing surface.

The results I will present are joint work with S. Alexakis and A. Nachman.

Zoom link: contact Peter McGrath host to get the link.