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{\bf \fontsize{14}{16pt}\selectfont
 INVISIBILITY AND INVERSE PROBLEMS
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Matti Lassas
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Helsinki University of Technology, Institute of Mathematics\\
P.O. Box 1100, FI-02015 TKK, Finland\\
email: matti.lassas@tkk.fi
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There has recently been considerable interest in the possibility, both
theoretical and practical, of invisibility of objects to different types of
waves.  We construct several examples of cloaking enclosures covered
with anisotropic materials. These examples have a close connection
to earlier works carried out for the case of conductivity equation
[1]. When the conductivity is not bounded from below, 
one can cover any object with a properly chosen
anisotropic material so that the covered object appears in all
boundary measurement similar to a domain with constant conductivity.
Clearly, this kind of counter example gives us theoretical
instructions how to cover
an object so that it appears ``invisible'' in zero frequency measurements.
In this talk we consider similar kind of result for all frequencies.

We review the results on the counter examples for Maxwell's and conductivity
equations and consider in detail the existence of the finite energy solutions.

The results have been done in collaboration with 
A. Greenleaf, Y. Kurylev and G. Uhlmann, and with
K. Astala and L. P\"aiv\"arinta.



References:

[1] A. Greenleaf, M. Lassas, G. Uhlmann: On nonuniqueness for Calderon's inverse problem,  Mathematical Research Letters 10 (2003), 685-693.

[2] 
A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Electromagnetic wormholes and virtual magnetic monopoles from metamaterials. Physical Review Letters. 99, 183901 (2007).

[3] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Full-wave invisibility of active devices at all frequencies, Communications in Mathematical Physics
275 (2007), 749-789.


[4] K. Astala, M. Lassas, L. P\"aiv\"arinta: Limits of visibility
 and invisibility in Calderon's problem, in preparation.

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