Seminar | March 19 | 4:10-5 p.m. | 740 Evans Hall
Mihajlo Cekic, Max Planck Institute for Mathematics (Bonn)
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. We sketch the proof of uniqueness up to gauges fixing the boundary for smooth line bundles. The proof uses new techniques, involving unique continuation principles for degenerate elliptic equations and an analysis of the zero set of solutions to an elliptic PDE. Time permitting, we will also discuss some (counter) examples concerning zero sets of determinants of matrix solutions to elliptic PDE.