Geometric configuration and Fourier analysis

这项建议的主旨是研究离散、连续和算术环境中的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确定所有可能的有限点构形中的相当大比例直到同余。这项研究涉及分析方法、数论方法和组合方法之间的紧密互动，在无缝共生中允许更深入地理解相关的技术和想法。该提案的技术和思想不仅涉及不同数学领域之间的相互作用。它们还涉及理论计算机科学的想法，潜在地应用于数据挖掘、编码、信号处理、生物信息学和现代科学的许多其他领域。现代调和分析是皮埃尔？S的主要专长领域，它是几乎所有科学学科都具有相关性的技术和思想的宝库，这种影响将在未来几年和几十年内继续增长。

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.