Discrepancy Theory and Analysis

差异理论与分析

• 批准号: 1101519
• 负责人: Dmitriy Bilyk
• 金额: $ 10.52万
• 依托单位: University South Carolina Research Foundation
• 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: 2011-07-01 至 2012-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101519&HistoricalAwards=false
• 关键词: Discrepancy; Theory; Analysis

Discrepancy theory studies various forms of the following question: How well can uniform distribution be approximated by a discrete set? This project is devoted to exploring fundamental problems and conjectures of discrepancy theory through the prism of harmonic and functional analysis. The strategic goals of the project are two-fold: to infuse discrepancy theory with new methods stemming from analysis, as well as to enrich the field of analysis with new problems and ideas. The program includes a wide range of problems and topics: understanding the precise behavior of the irregularity of distributions in various function spaces, producing new constructions of various low-discrepancy sets, investigating the correlation between discrepancy estimates and geometry, as well as numerous connections of the field to other areas of mathematics, such as probability and approximation theory.The ideas and questions of discrepancy theory are strongly interlaced with other areas of mathematics: combinatorics, geometry, number theory, probability, approximation theory. Moreover, this field is closely connected to computational mathematics, namely, the methods of numerical integration, and thus, it has direct applications in science, engineering, finance etc. The scientific output of this project will expand and deepen the understanding of the underlying phenomena of discrepancy theory and its relations to other branches of mathematics, as well as yield important applications to science and technology. The results will be broadly disseminated through publications in high-level journals, presentations in conferences, seminars, and colloquia, active collaborations with researchers around the world, a number of educational and mentoring activities.

差异理论研究以下问题的各种形式：离散集能在多大程度上近似均匀分布？该项目致力于通过调和和泛函分析的棱镜探索差异理论的基本问题和猜想。 该项目的战略目标有两个：将差异理论与分析产生的新方法相结合，以及用新问题和新想法丰富分析领域。该计划包括广泛的问题和主题：了解各种函数空间中分布不规则性的精确行为，产生各种低差异集的新构造，研究差异估计与几何之间的相关性，以及差异理论的思想和问题与其他数学领域密切相关：组合学、几何、数论、概率、逼近论。此外，该领域与计算数学（即数值积分方法）密切相关，因此在科学、工程、金融等领域有直接应用。该项目的科学成果将扩大和加深对潜在现象的理解差异理论及其与其他数学分支的关系，并在科学和技术中产生重要的应用。研究结果将通过高水平期刊上的出版物、会议、研讨会和座谈会中的演讲、与世界各地研究人员的积极合作以及一系列教育和指导活动来广泛传播。