Optimal Measures and Point Configurations: A Harmonic Analysis Approach

最佳测量和点配置：谐波分析方法

• 批准号: 2054606
• 负责人: Dmitriy Bilyk
• 金额: $ 26.59万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054606&HistoricalAwards=false
• 关键词: Optimal; Measures; Point; Configurations; Harmonic

Optimal point configurations and distributions arise in numerous areas in science applications including sampling of data, signal processing, equilibria of particles and charges, crystal formation, population distribution, discretization of surfaces, and coding theory. They are also extremely useful in numerical analysis and in fields that call for numerical computation of high dimensional integrals such as finance, often yielding better results than Monte Carlo methods. Studied within the framework of this project will be optimal point configurations and measures through the prism of harmonic analysis. The project will also study fundamental questions regarding uniform distribution of energy minimizers; determination of optimal distributions; discrete clustering phenomena; the interplay between geometry and uniform distribution; connections between discrepancy and energy minimization. In addition, the project will study related questions in applied harmonic analysis, compressed sensing, and frame theory. The results and topics of this project will be disseminated through publications and presentations, and young mathematicians will actively be mentored. The quality of a point distribution may be expressed or measured in a variety of different ways depending on the problem at hand. Examples include lattices, cubature formulas, codes, packings, and random point processes. This project is focused on two central inter-connected concepts, both of which quantify the uniformity of distributions: discrepancy and energy minimization. The former compares a given distribution with the uniform measure by looking at the errors on a certain class of test sets or test functions. The latter interprets points as “particles”, which interact according to a certain potential. While progress in these subjects often relies on methods and ideas of harmonic analysis, many of these connections have only been discovered recently. This leads to natural cross-fertilization: analysis provides the necessary tools, and at the same time it is enriched with new questions. The project will study several grand topics related to discrepancy and energy optimization including clustering of minimizers, attractive-repulsive energies and their relations to signal processing and applied harmonic analysis (tight frames, SIC-POVMs, mutually unbiased bases), discrete and metric geometry (equiangular lines, distance integrals), tessellations of spheres and their relation to discrepancy, one-bit compressed sensing, and embedding of metric spaces, direct interplay between energy and discrepancy and its applications to various questions and conjectures, the long standing open question of the exact asymptotics of discrepancy with respect to axis-parallel rectangles, as well as sibling questions in harmonic analysis, approximation and probability theory. Ubiquitous connections between the subjects bring together a variety of questions into an integral project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

最佳点配置和分布出现在科学应用的许多领域，包括数据采样、信号处理、粒子和电荷的平衡、晶体形成、布居分布、表面离散化和编码理论。它们在数值分析和金融等需要数值计算高维积分的领域也非常有用，通常比蒙特卡罗方法产生更好的结果。在这个项目的框架内，将通过调和分析的棱镜来研究最优化的点配置和措施。该项目还将研究关于能量最小化的均匀分布、最佳分布的确定、离散聚集现象、几何形状与均匀分布之间的相互作用、差异与能量最小化之间的联系等基本问题。此外，该项目还将研究应用谐波分析、压缩传感和框架理论中的相关问题。该项目的成果和主题将通过出版物和演示文稿传播，青年数学家将得到积极的指导。根据手头的问题，可以用各种不同的方式来表示或测量点分布的质量。例如格子、立方公式、编码、包装和随机点过程。这个项目的重点是两个相互关联的核心概念，这两个概念都量化了分布的一致性：差异和能量最小化。前者通过查看某一类测试集或测试函数上的误差，将给定的分布与均匀度量进行比较。后者将点解释为“粒子”，它们按照一定的势能相互作用。虽然这些学科的进展往往依赖于调和分析的方法和思想，但其中许多联系是最近才被发现的。这导致了自然的交叉受精：分析提供了必要的工具，同时也丰富了新的问题。该项目将研究与差异和能量优化有关的几个大主题，包括极小项的聚集，吸引-排斥能量及其与信号处理和应用调和分析的关系(紧框架，SIC-POVM，相互无偏的基)，离散和度量几何(等角线，距离积分)，球体的镶嵌及其与差异的关系，一比特压缩传感，度量空间的嵌入，能量和差异之间的直接相互作用及其在各种问题和猜想中的应用，关于轴-平行矩形的差异的精确渐近的长期悬而未决的问题，以及调和分析、近似和概率理论中的兄弟问题。主题之间无处不在的联系将各种问题汇聚到一个完整的项目中。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Optimal measures for $p$-frame energies on spheres

球体上 $p$ 框架能量的最佳测量

• DOI: 10.4171/rmi/1329
• 发表时间: 2022
• 期刊: Revista Matemática Iberoamericana
• 作者: Bilyk, Dmitriy;Glazyrin, Alexey;Matzke, Ryan;Park, Josiah;Vlasiuk, Oleksandr
• 通讯作者: Vlasiuk, Oleksandr

Positive definiteness and the Stolarsky invariance principle

正定性和斯托拉斯基不变性原理

• DOI: 10.1016/j.jmaa.2022.126220
• 发表时间: 2022
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Bilyk, Dmitriy;Matzke, Ryan W.;Vlasiuk, Oleksandr
• 通讯作者: Vlasiuk, Oleksandr

Potential theory with multivariate kernels

多元核势理论

• DOI: 10.1007/s00209-022-03000-z
• 发表时间: 2022
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Bilyk, Dmitriy;Ferizović, Damir;Glazyrin, Alexey;Matzke, Ryan W.;Park, Josiah;Vlasiuk, Oleksandr
• 通讯作者: Vlasiuk, Oleksandr