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.