# X-ray transforms on surfaces and degenerate elliptic operators

## Location

384H

Tuesday, January 24, 2023 4:00 PM

François Monard (UCSC)

**Abstract:**On a Riemannian manifold with boundary, the X-ray transform integrates a function

or a tensor field along all geodesics through the manifold. The reconstruction of the integrand of

interest from its X-ray transform is the basis of important inverse problems in seismology and medical imaging.

The inversion of the X-ray transform is often done by inverting the normal operator (composition

of the X-ray transform and its adjoint), and the appropriate spaces where such an operator can

be inverted, notably addressing the question of boundary behavior, are an object of recent study.

Such spaces include Frechet spaces of 'polyhomogeneous conormal' type, and non-standard

Sobolev scales (some are transmission spaces a la Hormander, others are modeled after degenerate

elliptic operator of Kimura/Keldysh type).

Along the way, new relations have been found between X-ray transforms on some simple surfaces,

and a special family of degenerate elliptic operators, and exploiting these relations requires a deeper

study of the latter operators, in particular finding boundary triples for them.

This talk will attempt to survey recent progress on these topics. This is based on recent joint works with

Rafe Mazzeo; Rohit Mishra and Joey Zou; Joey Zou.