American Institute of Mathematics, Palo Alto, California
Alexander Koldobsky, Dmitry Ryabogin, and Artem Zvavitch
This workshop, sponsored by AIM and the NSF, concerns the interface between convex geometry and harmonic analysis. Particular attention will be given to applications of Fourier analysis to the study of sections and projections of convex bodies.
The Fourier analytic approach to sections and projections of convex bodies has recently been developed and has led to several results including Fourier analytic solutions of the Busemann-Petty and Shephard problems, characterizations of intersection and projection bodies, extremal sections and projections of certain classes of bodies. The main idea of this approach is to express different properties of convex bodies in terms of the Fourier transform and then use methods of Fourier analysis to solve geometric problems.
One direction of the discussion will focus on the duality between sections and projections that remains one of the most intriguing mysteries of convex geometry. Many results on sections and projections are similar ("dual") to each other, but the proofs are usually very different. In many cases, the Fourier approach provides unified treatment of sections and projections, and it would be very interesting to explore the connections that are behind this phenomenon. Of particular interest is the question of whether the classes of intersection bodies and polar projection bodies are isomorphically equivalent.
Another direction is to try to extend the Fourier approach from isometric convex geometry to the asymptotic theory of convex bodies, where one is mostly interested in phenomena occurring when the dimension goes to infinity. This theory has numerous applications to functional analysis, probability, computer science.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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