Gaussian Curvature, Geometric Combinatorics and the Fourier Transform

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PI: Alex Iosevich, University of Missouri, ColumbiaDMS-0245369Abstract:The main theme of this proposal is interaction between analytic, combinatorial and number theoretic methods. The specific problems discussed in the proposal include the distribution of lattice points in convex domains, analysis and combinatorics of distance sets, decay properties ofthe Fourier transform of characteristic functions of bounded domains, the existence and non-existence of Fourier bases, and bounded-ness of maximal averages over hypersurfaces. Each category has an important geometric combinatorial component. In addition, the distribution of lattice points in convex domains, the problems involving distance sets, and the study of Fourier bases have interesting number theoretic aspects. Finally, these problems have an impact on each other. For example, decay properties of the Fourier transforms of characteristic functions of sets are used in the study of all the aforementioned problems. Similarly, the geometric combinatorial techniques developed in the study of the existence of Fourier bases have also been applied to the study of the distribution of lattice points and properties of distance sets.While many of the issues described in this proposal are quite theoretical, they are closely connected with widely applicable concepts and problems. Many of techniques used to study the existence of Fourier bases are strongly related to concepts used in data transmission and coding. The techniques developed in the study of maximal averages over surfaces have been appliedover the years to obtain regularity results for the linear and non-linear wave equations, highly useful in physics and engineering. It is quite interesting is that these results, in turn, have deep combinatorial applications, creating a cycle of useful and beautiful applications.