Elements of noncommutative geometry in inverse problems on manifolds

TL;DR

This paper introduces an algebraic approach using noncommutative geometry to solve inverse problems on manifolds, recovering the manifold from boundary measurements via C*-algebras and their spectra.

Contribution

It develops an algebraic version of the boundary control method, linking inverse boundary data to noncommutative geometric structures for manifold reconstruction.

Findings

Inverse data determine a C*-algebra associated with the manifold.

The spectrum of this algebra is topologically identical to the manifold.

The method provides a new algebraic framework for inverse boundary problems.

Abstract

We deal with two dynamical systems associated with a Riemannian manifold with boundary. The first one is a system governed by the scalar wave equation, the second is governed by the Maxwell equations. Both of the systems are controlled from the boundary. The inverse problem is to recover the manifold via the relevant measurements at the boundary (inverse data). We show that the inverse data determine a C*-algebras, whose (topologized) spectra are identical to the manifold. By this, to recover the manifold is to determine a proper algebra from the inverse data, find its spectrum, and provide the spectrum with a Riemannian structure. The paper develops an algebraic version of the boundary control method, which is an approach to inverse problems based on their relations to control theory.