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.