**
Below is the ascii version of the abstract for 04-129.
The html version should be ready soon.**Gaik Ambartsoumian and Peter Kuchment
On the injectivity of the circular Radon transform arising in thermoacoustic tomography
(53K, LATEX)
ABSTRACT. The circular Radon transform integrates a function over the set of
all spheres with a given set of centers. The problem of
injectivity of this transform (as well as inversion formulas,
range descriptions, etc.) arises in many fields from approximation
theory to integral geometry, to inverse problems for PDEs, and
recently to newly developing types of tomography. A major
breakthrough in the $2D$ case was made several years ago in a work
by M.~Agranovsky and E.~T.~Quinto. Their techniques involved
intricate microlocal analysis and knowledge of geometry of zeros
of harmonic polynomials in the plane, which are somewhat
restrictive in more general circumstances. Since then there has
been an active search for alternative methods, especially the ones
based on simple PDE techniques. The article discusses known and
provides new results that one can obtain by methods that
essentially involve only the finite speed of propagation and
domain dependence for the wave equation.