Calderon problem for Yang-Mills connections

In the classical Calderon conjecture we want to recover a metric on a compact manifold, up to diffeomorphism fixing the boundary, from the Dirichlet-to-Neumann (DN) map of the metric Laplacian. This problem is routinely motivated by applications and is open in dimension 3 and higher. In this talk, we fix the metric and consider the DN map of the connection Laplacian for Yang-Mills connections on vector bundles. We sketch the proof of uniqueness up to gauges fixing the boundary and propose two approaches. In the first one we develop a new technique, involving degenerate unique continuation principles and an analysis of the zero set of solutions to an elliptic PDE. The second argument involves a Runge-type approximation along curves to recover holonomy and we are able to show uniqueness of both an arbitrary bundle and a Yang-Mills connection.