Fourier Analytic Approach to the Geometric Tomography

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ABSTRACT:The main goal of this proposal is to study problems in Convex Geometry and Geometric Tomography using methods of Harmonic Analysis. Geometric Tomography is the area of mathematics where one investigates properties of solids based on the information about sections and projections of these solids. It borrows ideas and methods from many fields of Theoretical Mathematics, such as integral and differential geometry, statistics, and Fourier analysis. But perhaps its biggest overlap is with Convex Geometry. A major component of this proposal is to study problems which arise naturally from the recent work of the PI on Fourier analytic approach to the case of the most general measure of sections of star bodies. This method provided a number of new links between harmonic analysis, probability theory and convex geometry. PI intends to continue his work in this area and to consider a number of questions motivated by Slicing Problem and Isomorphic Busemann-Petty problem. PI also proposes a new approach to the study of the geometric properties of convex sets with respect to the Gaussian measure. As an application PI consider problems connected to the Gaussian Correlation Conjecture. The idea, of this method, is to restate the conjecture in the language of Geometric Tomography and to apply recently developed Fourier Analytic techniques. Geometric Tomography has a lot of real life applications. It has goals similar to those of many related and often practical areas. One of the best known examples is Classical Computer Aided Tomography, which aims to reconstruct the density of objects by means of their line integrals. Other examples are crystallography, robotics, stereology, and electron microscopy. One of the ideas of this proposal is to connect theoretical results from Convex Geometry to those practical areas via Geometric Tomography. Another component of the broader impact of proposed research lies in the training of graduate and undergraduate students. Indeed, a lot of problems addressed in this proposal stated in such a way that they are intuitively clear not only to graduate but to undergraduate students. On the other hand, the answers for many of those problems are quite counterintuitive which stimulates an interest of students to the subject and Mathematics in general!