Comparison Problems for Radon Transforms

April 1, 2024

4:00PM - 5:00PM

EA 160

Add to Calendar 2024-04-01 16:00:00 2024-04-01 17:00:00 Comparison Problems for Radon Transforms Title: Comparison Problems for Radon TransformsSpeaker: Michael Roysdon (Case Western Reserve University)Speaker's URL: https://sites.google.com/case.edu/roysdon-michael/homeAbstract: At the start of the 20th century J. Radon answered the following question: can one reconstruct a function based on its integral on lines? While this question is simple at its core, the methods involved have had a lasting effects in various scientific fields, and have even brought about two entire fields of mathematics: geometric tomography and analytic tomography. Each of these fields concern questions about determining information about of an object given lower-dimensional information; for example, information about the volume of the object knowing the volumes of sections and projections of that object onto planes. Inspired by the famed Busemann-Petty problem from convex geometry and geometric tomography (~1954), we address more general questions of this nature in the realm of analytic tomography. We ask the very simple question: given a pair of even, non-negative, continuous and integrable functions f and g, such that the Radon transform of f is pointwise smaller than the Radon transform of g, does it necessarily follow that the L^p-norm of f is smaller than the L^p norm of g when p >1? We address this question for two types of Radon transforms: the classical Radon transform and the spherical Radon transform. The solution to this question is quite subtle and requires techniques from Harmonic Analysis and Fourier Analysis. As it turns out, this question is intimately related to the slicing problem of Bourgain (a question from Asymptotic Geometric Analysis (Geometric Probability)), reverse estimates for the Radon transform due to Oberlin and Stein from the 1980s, and finally some very recent estimate on the Radon transform due to Bennett and Tao. If time permits, we will discuss a lower-dimensional analogue of this problem. Based on a joint work with Alexander Koldobsky and Artem Zvavitch.URL associated with Seminar: https://u.osu.edu/hascv/ EA 160 OSU ASC Drupal 8 ascwebservices@osu.edu America/New_York public

Add to Calendar 2024-04-01 16:00:00 2024-04-01 17:00:00 Comparison Problems for Radon Transforms Title:  Comparison Problems for Radon TransformsSpeaker:  Michael Roysdon (Case Western Reserve University)Speaker's URL:  https://sites.google.com/case.edu/roysdon-michael/homeAbstract:  At the start of the 20th century J. Radon answered the following question: can one reconstruct a function based on its integral on lines? While this question is simple at its core, the methods involved have had a lasting effects in various scientific fields, and have even brought about two entire fields of mathematics: geometric tomography and analytic tomography. Each of these fields concern questions about determining information about of an object given lower-dimensional information; for example, information about the volume of the object knowing the volumes of sections and projections of that object onto planes. Inspired by the famed Busemann-Petty problem from convex geometry and geometric tomography (~1954), we address more general questions of this nature in the realm of analytic tomography. We ask the very simple question: given a pair of even, non-negative, continuous and integrable functions f and g, such that the Radon transform of f is pointwise smaller than the Radon transform of g, does it necessarily follow that the L^p-norm of f is smaller than the L^p norm of g when p >1? We address this question for two types of Radon transforms: the classical Radon transform and the spherical Radon transform. The solution to this question is quite subtle and requires techniques from Harmonic Analysis and Fourier Analysis. As it turns out, this question is intimately related to the slicing problem of Bourgain (a question from Asymptotic Geometric Analysis (Geometric Probability)), reverse estimates for the Radon transform due to Oberlin and Stein from the 1980s, and finally some very recent estimate on the Radon transform due to Bennett and Tao. If time permits, we will discuss a lower-dimensional analogue of this problem. Based on a joint work with Alexander Koldobsky and Artem Zvavitch.URL associated with Seminar:  https://u.osu.edu/hascv/ EA 160 Department of Mathematics math@osu.edu America/New_York public

Title: Comparison Problems for Radon Transforms

Speaker: Michael Roysdon (Case Western Reserve University)

Speaker's URL: https://sites.google.com/case.edu/roysdon-michael/home

Abstract: At the start of the 20th century J. Radon answered the following question: can one reconstruct a function based on its integral on lines? While this question is simple at its core, the methods involved have had a lasting effects in various scientific fields, and have even brought about two entire fields of mathematics: geometric tomography and analytic tomography. Each of these fields concern questions about determining information about of an object given lower-dimensional information; for example, information about the volume of the object knowing the volumes of sections and projections of that object onto planes.

Inspired by the famed Busemann-Petty problem from convex geometry and geometric tomography (~1954), we address more general questions of this nature in the realm of analytic tomography. We ask the very simple question: given a pair of even, non-negative, continuous and integrable functions f and g, such that the Radon transform of f is pointwise smaller than the Radon transform of g, does it necessarily follow that the L^p-norm of f is smaller than the L^p norm of g when p >1? We address this question for two types of Radon transforms: the classical Radon transform and the spherical Radon transform. The solution to this question is quite subtle and requires techniques from Harmonic Analysis and Fourier Analysis. As it turns out, this question is intimately related to the slicing problem of Bourgain (a question from Asymptotic Geometric Analysis (Geometric Probability)), reverse estimates for the Radon transform due to Oberlin and Stein from the 1980s, and finally some very recent estimate on the Radon transform due to Bennett and Tao. If time permits, we will discuss a lower-dimensional analogue of this problem.

Based on a joint work with Alexander Koldobsky and Artem Zvavitch.

URL associated with Seminar: https://u.osu.edu/hascv/

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Seminar - Harmonic Analysis and Several Complex Variables