BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Yuzhou (Joey) Zou (UC Santa Cruz) DTSTART:20220421T160000Z DTEND:20220421T170000Z DTSTAMP:20260423T021147Z UID:Inverse/81 DESCRIPTION:Title: The $C^\\infty$-isomorphism property for a class of singularly-weighted X -ray transforms\nby Yuzhou (Joey) Zou (UC Santa Cruz) as part of Inter national Zoom Inverse Problems Seminar\, UC Irvine\n\n\nAbstract\nWe consi der the mapping properties of singularly-weighted normal operators associa ted to X-ray transforms on manifolds with boundary. While normal operators associated to geodesic X-ray transforms in "simple" settings are known to be elliptic pseudodifferential operators in the interior\, their behavior near the boundary is more subtle\; in particular the normal operators nee d to be precomposed with weights in order to even map $C^\\infty$ of a man ifold with boundary back to itself. This motivates asking which choice of weights guarantee the normal operator to be an isomorphism of $C^\\infty$\ ; such questions arise in considering theoretical guarantees for the consi stency and uncertainty quantification of statistical recovery algorithms\, where one needs to know on what spaces the operator can be considered inv ertible. In this talk\, we will show that a particular family of weights o n the Euclidean disk and on simple disks of constant curvature do give ris e to normal operators which are isomorphisms on $C^\\infty$. The proof inv olves deriving the Singular Value Decomposition of a weighted X-ray transf orm and studying certain function spaces based on the singular vectors of the X-ray transform\, which coincides with the eigenfunctions of a particu lar degenerately elliptic Kimura-type differential operator. Joint work wi th Rohit Kumar Mishra and Francois Monard.\n LOCATION:https://researchseminars.org/talk/Inverse/81/ END:VEVENT END:VCALENDAR