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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

Invisibility and inverse problems


Authors: Allan Greenleaf, Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann
Journal: Bull. Amer. Math. Soc. 46 (2009), 55-97
MSC (2000): Primary 35R30, 78A46; Secondary 58J05, 78A10
Published electronically: October 14, 2008
MathSciNet review: 2457072
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Abstract: We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws under changes of variables which allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something is being hidden; the object, which casts no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other devices having striking effects on wave propagation, unseen in nature, are also possible. These designs are initially based on the transformation laws of the relevant PDEs, but due to the singular transformations needed for the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved.


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Additional Information

Allan Greenleaf
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

Yaroslav Kurylev
Affiliation: Department of Mathematics, University College London, Gower Street, London, WC1E 5BT, United Kingdom

Matti Lassas
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, Helsinki 02015, Finland

Gunther Uhlmann
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

DOI: http://dx.doi.org/10.1090/S0273-0979-08-01232-9
PII: S 0273-0979(08)01232-9
Received by editor(s): June 30, 2008
Published electronically: October 14, 2008
Additional Notes: The first author was partially supported by NSF grant DMS-0551894.
The third author was partially supported by Academy of Finland CoE Project 213476.
The fourth author was partially supported by the NSF and a Walker Family Endowed Professorship.
Article copyright: © Copyright 2008 American Mathematical Society