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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Stability of the geodesic ray transform in the pre
sence of caustics - Sean Holman\, Manchester
DTSTART;TZID=Europe/London:20161109T160000
DTEND;TZID=Europe/London:20161109T170000
UID:TALK66505AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/66505
DESCRIPTION:Given a compact Riemannian manifold $(M\,g)$ with
boundary\, the geodesic ray transform is the mappi
ng which takes a function on $M$ to its integrals
over the maximally extended geodesics of $(M\,g)$.
We are interested primarily in two questions: whe
ther this transform is injective\, and whether the
re is a stability estimate between appropriate Sob
olev spaces for its inversion. It is well known th
at for so-called “simple manifolds”\, which in par
ticular do not have caustics\, the transform is in
jective\, and there is a stability estimate. On th
e 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 p
ossible. This is the case even though there are tw
o dimensional examples which have caustics\, but f
or which the transform is injective. The question
motivating this talk is whether the same phenomeno
n happens in three dimensions. The talk will exami
ne recent results on the stability of the inversio
n of the geodesic ray transform in the presence of
caustics in three dimensions\, contrasting them w
ith what is known on the injectivity.
LOCATION:MR13
CONTACT:Ivan Smith
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