## 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.