Noncommutative geometry and the tomography of manifolds

Abstract

The tomography of manifolds describes a range of inverse problems in which we seek to reconstruct a Riemannian manifold from its boundary data (the “Dirichlet–Neumann” mapping, the reaction operator, and others). Different types of data correspond to physically different situations: the manifold is probed by electric currents or by acoustic or electromagnetic waves. In our paper we suggest a unified approach to these problems, using the ideas of noncommutative geometry. Within the framework of this approach, the underlying manifold for the reconstruction is obtained as the spectrum of an adequate Banach algebra determined by the boundary data.