Uniform Distribution and Harmonic Analysis

本研究项目研究的是用离散对象逼近连续对象的基本问题，并对过程中不可避免地产生的误差进行评估。这类问题出现在大多数数学分支中：概率论、近似论、数论、组合学和离散几何，以及任何需要计算多元积分的科学、工程或金融学科。最近发现了一些不那么明显但更深刻的联系：一些已经被形式化和理解，而另一些只是启发式的，等待进一步的研究。均匀分布理论是这个项目的重点，几乎每一个关键的发展都利用了实数分析、泛函分析，特别是调和分析的应用。与此同时，许多利用均匀分布理论的其他领域的专家历来忽视的许多现代调和分析方法才刚刚开始在各自的领域站稳脚跟。许多对均匀分布理论至关重要的问题，特别是在更高维度上的问题，仍然是悬而未决的。该项目研究的问题包括：高维最优偏差的精确渐近，一位压缩感知，度量空间的嵌入，偏差与能量最小化之间的相互作用，均匀分布点集的构造(低偏差，立方体，能量最小化，晶格)，函数空间中的偏差估计和数值积分，探索几何对均匀分布性质的影响，以及离散几何问题(镶嵌，覆盖，填充)。这些问题的进展必须与分析和其他领域的重要问题和猜想的进展紧密交织在一起，将一系列明显互不相关的问题和主题粘合在一起。该项目的结果预计将影响数学的几个领域，用新的结果、想法和方法丰富和交叉培养它们。

This research project deals with the fundamental problem of approximating continuous objects by discrete ones and evaluating the errors that inevitably arise in the process. Questions of this kind come up in most branches of mathematics: probability, approximation theory, number theory, combinatorics, and discrete geometry, as well as in any discipline of science, engineering, or finance that demands computations of multivariate integrals. A number of less obvious and more profound connections have been discovered recently: some have already been formalized and understood, while others are only heuristic and await further research. Almost every pivotal development in uniform distribution theory, the focus of this project, has exploited applications of real analysis, functional analysis, and especially harmonic analysis. At the same time, numerous modern methods of harmonic analysis that were historically overlooked by experts in other fields that make use of uniform distribution theory are only starting to gain footholds in their areas. Many questions of utmost importance to uniform distribution theory, especially in higher dimensions, remain wide open. The questions under investigation in this project include the longstanding problem of precise asymptotics of optimal discrepancy in higher dimensions, one-bit compressed sensing, embeddings of metric spaces, interplay between discrepancy and energy minimization, constructions of well-distributed point sets (low-discrepancy, cubature, energy-minimizing, lattices), discrepancy estimates and numerical integration in function spaces, exploring the effect of geometry on uniform distribution properties, and problems of discrete geometry (tessellations, coverings, packings). Progress on these questions has to be tightly intertwined with advances on important problems and conjectures in analysis and other fields, gluing together a mosaic of apparently disconnected questions and topics. The outcomes of the project are expected to impact several areas of mathematics, enriching and cross-fertilizing them with new results, ideas, and methods.