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SUMMARY:Stability of the geodesic ray transform in the presence of caustic
 s - Sean Holman\, Manchester
DTSTART:20161109T160000Z
DTEND:20161109T170000Z
UID:TALK66505@talks.cam.ac.uk
CONTACT:Ivan Smith
DESCRIPTION:Given a compact Riemannian manifold $(M\,g)$ with boundary\, t
 he geodesic ray transform is the mapping which takes a function on $M$ to 
 its integrals over the maximally extended geodesics of $(M\,g)$. We are in
 terested primarily in two questions: whether this transform is injective\,
  and whether there is a stability estimate between appropriate Sobolev spa
 ces for its inversion. It is well known that for so-called “simple manif
 olds”\, which in particular do not have caustics\, the transform is inje
 ctive\, and there is a stability estimate. On the other hand\, in the two 
 dimensional case it has been proven that as soon as there are caustics no 
 stability estimate between any Sobolev spaces is possible. This is the cas
 e even though there are two dimensional examples which have caustics\, but
  for which the transform is injective. The question motivating this talk i
 s whether the same phenomenon happens in three dimensions. The talk will e
 xamine recent results on the stability of the inversion of the geodesic ra
 y transform in the presence of caustics in three dimensions\, contrasting 
 them with what is known on the injectivity.
LOCATION:MR13
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