Optimal Measures and Point Configurations: A Harmonic Analysis Approach

最佳点的配置和分布出现在许多科学应用领域，包括数据采样、信号处理、粒子和电荷的平衡、晶体形成、种群分布、表面离散化和编码理论。它们在数值分析和需要高维积分数值计算的领域（如金融）中也非常有用，通常比蒙特卡罗方法产生更好的结果。本项目将在研究框架内通过棱镜谐波分析优化点的配置和措施。该项目还将研究有关能源最小化器均匀分布的基本问题；确定最优分布；离散聚类现象；几何与均匀分布的相互作用；差异与能量最小化之间的联系。此外，本项目将研究应用谐波分析、压缩感知和框架理论的相关问题。该项目的成果和主题将通过出版物和演讲传播，并将积极指导年轻数学家。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。例子包括格、培养公式、代码、包装和随机点过程。该项目侧重于两个相互关联的核心概念，这两个概念都量化了分布的均匀性：差异和能量最小化。前者通过观察某一类测试集或测试函数上的误差，将给定分布与统一度量进行比较。后者将点解释为“粒子”，它们根据一定的势相互作用。虽然这些学科的进展往往依赖于谐波分析的方法和思想，但其中许多联系是最近才发现的。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相关的几个重大主题，包括最小化聚类、吸引-排斥能量及其与信号处理和应用谐波分析（紧框架、sic - povm、互无偏基）的关系、离散和度量几何（等角线、距离积分）、球体的细分及其与差异的关系、比特压缩感知和度量空间的嵌入。能量与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确切渐近性的长期开放问题，以及谐波分析，近似和概率论中的兄弟问题。学科之间无处不在的联系将各种各样的问题汇集成一个完整的项目。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.