Noncommutative geometry and the tomography of manifolds
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M. I. Belishev, M. N. Demchenko and A. N. Popov
Translated by: Christopher Hollings - Trans. Moscow Math. Soc. 2014, 133-149
- DOI: https://doi.org/10.1090/S0077-1554-2014-00239-9
- Published electronically: November 6, 2014
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.References
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Bibliographic 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
- 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.
- © Copyright 2014 by the authors
- 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
- MathSciNet review: 3308606