An inverse problem for fractional generalized connection Laplacians

Abstract: Given a Hermitian vector bundle over a closed Riemannian manifold, one can consider the fractional operator of $P^s$, $0<s<1$, associated with the generalized connection Laplacian $P:=\nabla^*\nabla+A$. After reviewing this construction and previous inverse problem results related to non-fractional operators, we present new work on recovering the global bundle structure, connection $\nabla$, potential $A$, and Riemannian metric from local source-to-solution data associated with the equation $P^su=f$ over an open set $U$. We conclude by discussing possible future avenues of research.