Manchester Geometry Seminar 2013/2014

Given a Riemannian manifold $(M,g)$ with boundary, the geodesic X-ray transform is the mapping $\mathcal{X}$ which takes a function on $M$ to its integrals along the geodesics between boundary points of $M$. We are interested in whether this map is injective, and whether the inverse is continuous. One approach is to begin by constructing a parametrix for the so-called normal operator $N = \mathcal{X}^t \circ \mathcal{X}$. 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 so 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 canonical relations can be given in terms of the geometry of the conjugate points in $M$. Along the way we also examine some geometric properties of conjugate points of Riemannian manifolds in general.