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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

   
 

 

Noncommutative geometry and the tomography of manifolds


Authors: M. I. Belishev, M. N. Demchenko and A. N. Popov
Translated by: Christopher Hollings
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2014, 133-149
MSC (2010): Primary 35R30, 46L60, 58B34, 93B28, 35Q61
DOI: https://doi.org/10.1090/S0077-1554-2014-00239-9
Published electronically: November 6, 2014
MathSciNet review: 3308606
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Abstract | References | Similar Articles | Additional Information

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.


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Additional Information

M. I. Belishev
Affiliation: Saint Petersburg Department of the Steklov Institute of Mathematics of the Russian Academy of Sciences, and Saint Petersburg State University
Email: belishev@pdmi.ras.ru

M. N. Demchenko
Affiliation: Saint Petersburg Department of the Steklov Institute of Mathematics of the Russian Academy of Sciences, and Saint Petersburg State University
Email: demchenko@pdmi.ras.ru

A. N. Popov
Affiliation: Saint Petersburg State University
Email: al_nik_popov@rambler.ru

DOI: https://doi.org/10.1090/S0077-1554-2014-00239-9
Keywords: Restoration of a Riemannian manifold from boundary data, impedance, acoustic and electromagnetic tomography, connections of tomography problems with noncommutative geometry
Published electronically: November 6, 2014
Additional Notes: The first author was supported by grants 14-01-00535A and NSh-1771.2014.1 from the Russian Foundation for Basic Research (RFFI), the second by grants 14-01-31388-mol-a and NSh-1771.2014.1 from the same source, and the third by grants 6.38.670.2013 and NSh-1292.2014.1 from Saint Petersburg State University.
Article copyright: © Copyright 2014 by the authors