Applications of Fourier analysis to convex geometry

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This mathematics research project by Aleksandr Koldobsky concerns the study of geometric properties of convex bodies based on information about sections and projections of these bodies. This research area has important applications to many areas of mathematics and science. Based on methods of Fourier analysis, Koldobsky has recently developed a new approach to the study of sections and projections of convex bodies. The idea is to express certain geometric characteristics of a body in terms of the Fourier transform and then use methods of harmonic analysis to solve geometric problems. This approach has led to several results including Fourier analytic solutions of the Busemann-Petty and Shephard problems,characterizations of intersection and projection bodies. Koldobsky will apply the Fourier approach to other open problems in the area, with the emphasis on newly discovered stability and separation properties in volume comparison problems and their applications to the hyperplane conjecture. Another direction is to study the duality between section and projections, in particular to establish isomorphic equivalence of intersection and polar projection bodies. An old problem in probability theory is to characterize all random vectors having the property that all linear combinations of coordinates have the same distribution, up to a constant. The classical examples are stable random vectors. Koldobsky will characterize all random vectors with this property.While the problems considered in this mathematics research project by Aleksandr Koldobsky are related to three different areas of mathematics (convex geometry, functional analysis and probability) the strategy of solution is common for most of the results - the questions are translated into the language of the Fourier transform and then treated as problems in harmonic analysis. In convex geometry, Koldobsky considers the problem of reconstructing a solid using data about plane sections and projections of this solid. The results and methods extend the classical techniques of x-ray tomography to situations where only special sets of sections or projections are available. In probability, Koldobsky studies stable processes, which are probabilistic laws that inherit the main self-reproductive property of the normal law. The underlying Fourier techniques developed by Koldobsky have led to important applications to other disciplines (e.g., neural networks in image processing; for example, one can find the main frequencies of a signal using only information about the maximal amplitude of this signal on intervals of time of equal lengths). An important part of the project is the involvement and training of graduate students and postdocs.