On an inverse problem for fractional connection Laplacians

Abstract

Classical inverse problems seek to determine the unknown coefficients of a PDE from boundary or local measurements of solutions. In the past few years, there has been a sharp increase in attention paid to inverse problems for fractional Laplacians and their associated nonlocal equations. While most of this research takes place on $\mathbb{R}^n$, recently [FGKU21] showed that the Riemannian metric on a closed manifold is uniquely determined by local Riemannian structure and a source-to-solution map for the fractional Laplace-Beltrami operator. Our paper [Chi22] generalizes this result by considering instead a fractional operator $P^s$, $0<s<1$, for connection Laplacian $P:=\nabla^*\nabla+A$ on a smooth Hermitian vector bundle over a closed, connected Riemannian manifold of dimension $n\geq 2$. Assuming local knowledge of the metric, Hermitian bundle, connection, potential, and source-to-solution map associated with $P^s$, we show that all of these geometric structures are determined globally up to gauge invariance and isometry. This thesis shares content with our preprint [Chi22], which is currently under revision. New additions include a more detailed discussion on fractional inverse problems and a significantly expanded exposition of some of the more technical tools involved.