Geometric configuration and Fourier analysis

这个建议的主要推力是研究离散，连续和算术设置中的k点配置。基本问题是确定给定向量空间的子集需要多大才能确保它决定所有可能的有限点构型的相当大的比例直到同余。该研究涉及分析，数论和组合方法之间的紧密互动，在无缝的共生关系中，允许对相关技术和思想的更深层次的理解。该提案的技术和思想不仅涉及数学不同领域之间的相互作用。它们还涉及理论计算机科学的思想，这些思想在数据挖掘、编码、信号处理、生物信息学和许多其他现代科学领域具有潜在的应用。现代谐波分析，什么是PI？它是一个技术和思想的宝库，在几乎所有科学学科中都找到了相关性，这种影响将在未来的几年和几十年里继续增长。

The main thrust of this proposal is the study of k-point configurations in discrete, continuous and arithmetic settings. The basic question is to determine how large a subset of a given vector space needs to be to ensure that it determines a substantial proportion of all possible finite point configurations up to congruence. The research involves a tight interaction between analytic, number theoretic and combinatorial methods in a seamless symbiosis that allows for a deeper understanding of the associated techniques and ideas. The techniques and ideas of the proposal do not only involve interaction between different areas of mathematics. They also involve ideas from theoretical computer science with potential applications to data mining, coding, signal processing, bioinformatics and many other areas of modern science. Modern harmonic analysis, which is PI?s main area of expertise, is a treasure trove of techniques and ideas that have found relevance in virtually every scientific disciplines and this influence will continue to grow in the years and decades to come.