Expanders, Fourier analysis and point-curve incidence theory

扩展器、傅里叶分析和点曲线关联理论

• 批准号: 1242660
• 负责人: Derrick Hart
• 金额: $ 6.96万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-12-07 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1242660&HistoricalAwards=false
• 关键词: Expanders; Fourier; analysis; point; curve

The main goal of the proposed project is to study the effect that the ambient field has on the behavior of set expanders. This is explored by studying the Euclidean analogs of two problems which were recently investigated in the context of vector spaces over finite fields. The first problem is a natural generalization of the Falconer distance problem to the study of point configurations. The second problem is drawn from a large class of problems known as the sum-product problems. The continuous nature of the first problem and the discrete nature of the second allows the researcher to incorporate a variety of methods from various fields in order to better illustrate the general principles involved. These include methods drawn from arithmetic and geometric combinatorics, number theory, and classical harmonic analysis. The juxtaposition of the methods from the Euclidean cases with the methods in the context of the finite field geometry could provide a more complete picture of the set expansion phenomenon.This project involves studying the interaction between several areas of mathematics using a problem-solving approach, in which two problems are considered whose analogs have known solutions in one subfield of mathematics known as finite field geometry. The problems are examined in two additional related subfields known as discrete and continuous Euclidean geometry. The goal of the project is to not only find solutions to the two problems in these alternate contexts, but to explore the interplay between a variety of known methods and techniques in order to obtain a more complete understanding of the scope of the general principles involved. The major areas of mathematics these problems are taken from include additive and geometric combinatorics, number theory, and classical harmonic analysis. These areas have a variety of applications in encryption, data mining, signal processing, and bioinformatics.

该项目的主要目标是研究环境场对集合扩展器行为的影响。 这是探讨通过研究欧几里得类似物的两个问题，最近调查的背景下，向量空间在有限领域。第一个问题是一个自然的推广的法尔科纳距离问题的研究点配置。 第二个问题是从一个大类的问题称为和产品的问题。 第一个问题的连续性和第二个问题的离散性使研究人员能够结合来自各个领域的各种方法，以便更好地说明所涉及的一般原理。 这些方法包括从算术和几何组合学，数论，和经典的谐波分析。 并置的方法，从欧几里德的情况下，在有限域几何的背景下的方法可以提供一个更完整的图片集expansion phenomenon.This project involves studying the interaction between several areas of mathematics using a problem solving approach，其中两个问题被认为是其类似物有已知的解决方案，在一个子领域的数学称为有限域几何。 这些问题在另外两个相关的子领域被称为离散和连续欧几里得几何检查。 该项目的目标不仅是在这些交替的背景下找到这两个问题的解决方案，而且要探索各种已知方法和技术之间的相互作用，以便更全面地了解所涉及的一般原则的范围。 这些问题的主要数学领域包括加法和几何组合学、数论和经典调和分析。 这些领域在加密、数据挖掘、信号处理和生物信息学中有各种应用。