ADGIS @ Aveiro
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Analysis and Differential Geometry International Seminar
University of Aveiro, Aveiro, Portugal
Aims and goals ZOOM
The online seminar Analysis and Differential Geometry International Seminar (ADGIS@Aveiro) is jointly organized by the research groups CHAG and OGTCG, as well as by the thematic line PICS. It aims to bring together specialists from analysis and geometry with particular emphasis on partial differential equations, differential geometry, analysis on filtered manifolds, singular (differential) operators, spectral theory, and their connections to other fields. ZOOM
The seminar will be accessible via the platform ZOOM. If you are interested in participate / attend please contact the organizers in order to receive the access code/link.
Seminar
May 25, at 14:00 (15:00 CEST), Steven Flynn, University of Bath, U.K.
Title: The Heisenberg X-ray transform: A first approximation Inverse Problems on sub-Riemannian manifolds
Abstract: The classical X-ray Transform maps a function on Euclidean space to a function on the space of lines on thisZEuclidean space by integrating the function over the given line.ZInverting the X-ray transform has wide-ranging applications, including to medical imaging and seismology.ZMuch work has been done to understand this inverse problem in Euclidean space, Euclidean domains, and more generally, for symmetric spaces andZRiemannian manifolds with boundary where the lines become geodesics. We formulate a sub-Riemannian version of the X-ray transform on the simplest sub-Riemannnian manifold, the Heisenberg group.ZHere serious geometric obstructions to classical inverse problems, such as existence of conjugate points, appear generically.ZWith tools adapted to the geometry, such as an operator-valued Fourier SliceZTheorem, we prove that an integrable function on the Heisenberg group is indeed determined by its line integrals over sub-Riemannian (as well as over its compatible Riemannian and Lorentzian) geodesics. ZOOM
We also pose an abundance of accessible follow-upZquestions, standard in the inverse problems community, concerning the sub-Riemannian case, and report progress answering some of them. ZOOM
This online seminar is supported by CIDMA - Center for Research and Development in Mathematics and Applications, and FCT - Fundação para a Ciência e a Tecnologia with references UIDB/04106/2020 and UIDP/04106/2020.
