Gaussian Curvature, Geometric Combinatorics and the Fourier Transform

Pi：Alex Iosevich，密苏里大学，哥伦比亚DMS-0245369摘要：这项建议的主要主题是分析、组合和数论方法之间的相互作用。文中讨论的具体问题包括：凸域上格点的分布、距离集的分析与组合、有界域上特征函数的傅里叶变换的衰减性、傅立叶基的存在与不存在以及超曲面上极大平均值的有界性。每个范畴都有一个重要的几何组合成分。此外，凸域上格点的分布，涉及距离集的问题，以及傅立叶基的研究都有有趣的数论方面。最后，这些问题是相互影响的。例如，集合的特征函数的傅里叶变换的衰减性被用于上述所有问题的研究。同样，在研究傅立叶基的存在性时发展起来的几何组合技术也被应用到格点的分布和距离集的性质的研究中。虽然本方案中描述的许多问题都是非常理论的，但它们与广泛适用的概念和问题密切相关。许多用于研究傅立叶基的存在的技术与数据传输和编码中使用的概念密切相关。多年来，在研究表面上的最大平均值方面发展起来的技术已被应用于获得线性和非线性波动方程的正则性结果，这在物理和工程上都很有用。非常有趣的是，这些结果反过来具有深刻的组合应用，创造了一个有用和美丽的应用循环。

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.