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.2010 Mathematics Subject Classification. Primary 35R30, 46L60, 58B34, 93B28, 35Q61. Key words and phrases. Restoration of a Riemannian manifold from boundary data, impedance, acoustic and electromagnetic tomography, connections of tomography problems with noncommutative geometry.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 134 M. I. BELISHEV, M. N. DEMCHENKO, AND A. N. POPOVmeasurements are formalised by the assignment of a reaction operator which transforms displacements of the boundary (functions of the boundary points and time) into forces on the boundary (functions of the same variables). The inverse problem consists of reconstructing the shell by means of the reaction operator. This situation also makes sense in the multidimensional case and has significant applications in three dimensions (defectoscopy, geophysics, ultrasound diagnostics in medicine and others).Electromagnetic tomography (EMT). A region in curved space (a three-dimensional Riemannian manifold with boundary) is illuminated by electromagnetic waves initiated by boundary sources. The waves interact with the internal structure and carry information on it to the boundary. The observer seeks to recover the structure: the topology and the metric.The principal difference between these similar situations and the traditional coefficient inverse problem consists of the following. In the latter, the support of the required coefficients (the domain space or the manifold; in IT, the shell) is assumed to be known: we have the ability to fix a point of the support and consider how to recover the unknown function at it (the conductivity of the shell, the density of the medium, the speed of the wave, etc.). In our settings, the support itself is subject to definition. An additional complication is that the correspondence "manifold → reaction operator" is not injective: two isometric manifolds with common boundary have identical reaction operators. For the observer, such manifolds respond identically to external stimuli; in principle, they are indistinguishable and it is unclear which of them to recover.

New notions and constructions of the boundary control method

“…As a consequence, the Maxwellian eikonal algebra E M is noncommutative. Nevertheless, the factor ĖM := E M / C ∞ over the ideal of compact operators turns out to be commutative and isometric to C(Ω) (M.N.Demchenko [13,14]), and finally we reach ĖM ∼ = Ω.…”

Eikonal Algebra on a Graph of Simple Structure

“…• As was mentioned above (see (7)), the waves propagate with the unit speed. As a consequence, for f ∈ F T γ one has…”

“…We refer the reader to the papers by S.Avdonin, P.Kuchment, P.Kurasov, V.Yurko and others. Eikonal algebras are dedicated to solving inverse problems [5,6,7]. However, in the given paper we do not solve inverse problems but study an algebra closely related to them.…”

“…These algebras are used for reconstruction of Riemannian manifolds via dynamical and/or spectral boundary inverse data. Namely, these data determine the relevant eikonal algebra, which is a commutative C*-algebra, whereas its spectrum (a set of irreducible representations) provides an isometric copy of the manifold under reconstruction and, thus, solves the problem [5,6,7].…”