Microlocal analysis of the geodesic X-ray transform

Sean Holman (University of Manchester)

Frank Adams 2, School of Mathematics, Alan Turing Building,

Given a Riemannian manifold (M,g) the geodesic X-ray transform is
the mapping X which takes a function on M to it’s integrals along the
geodesics of g. We are interested in whether this map is injective, and
whether inversion can be done in a stable manner. In the case that M is R2
and g is the Euclidean metric this problem corresponds with X-ray CT and
answers to these questions have been known for close to 100 years. When M
represents the interior of the Earth and g is the travel-time metric for
seismic waves this is the linearization of the travel time tomography
problem and is widely used in seismology. In this case there is no full
answer to these questions.

One approach is to begin by constructing a parametrix for the so-called
normal operator N = X t ◦ X . Such a construction shows that the problem of
inverting N is Fredholm and therefore only has a finite dimensional kernel
and the inversion is stable on a complement of that kernel. In the absence
of conjugate points N is known to be an elliptic pseudodifferential operator
of order −1 and the parametrix construction is standard, but when there are
conjugate points the situation is more complicated. This talk will present
new results showing that under certain hypotheses the operator N is equal to
a pseudodifferential operator plus some Fourier integral operators whose
geometric properties can be characterised in terms of the conjugate points
of the metric. I will also provide a bit of background on Fourier integral
operators.

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