Discrepancy Theory and Analysis

差异理论研究以下问题的各种形式：离散集如何很好地近似均匀分布？本项目致力于通过调和分析和泛函分析的棱镜来探索差异理论的基本问题和猜想。该项目的战略目标是双重的：用分析产生的新方法注入差异理论，以及用新的问题和想法丰富分析领域。该计划包括广泛的问题和主题：理解各种函数空间中分布的不规则性的精确行为，产生各种低差异集的新结构，调查差异估计与几何之间的相关性，以及该领域与其他数学领域的众多联系，例如概率和近似理论。差异理论的思想和问题与其他数学领域紧密交织：组合学、几何、数论、概率论、近似理论。此外，该领域与计算数学，即数值积分方法密切相关，因此，它在科学、工程、金融等领域都有直接的应用。该项目的科学成果将扩大和加深对差异理论的基本现象及其与其他数学分支的关系的理解，并在科学和技术方面产生重要应用。研究结果将通过在高级别期刊上的出版物、在会议、研讨会和座谈会上的发言、与世界各地研究人员的积极合作以及一些教育和指导活动广泛传播。

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.