Elements of noncommutative geometry in inverse problems on manifolds

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.

References in corpus (5)

• An Introduction to Noncommutative Spaces and their Geometry
• Reconstruction of manifolds in noncommutative geometry
• On the spectral characterization of manifolds
• C*-algebras and inverse problem of electrodynamics
• Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements

Cited by in corpus (4)

• Wave model of the Sturm-Liouville operator on the half-line
• Eikonal algebra on a graph of simple structure
• Canonical form of $C^{\ast }$-algebra of eikonals related to the metric graph
• On the Cauchy problem for the wave equation with data on the boundary