Uniform Distribution and Harmonic Analysis

本研究项目涉及用离散对象近似连续对象的基本问题，并评估在此过程中不可避免地出现的误差。这类问题出现在数学的大多数分支中：概率论、逼近论、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。 最近发现了一些不太明显但更深刻的联系：一些已经被正式确定和理解，而另一些只是启发性的，有待进一步研究。 几乎每一个关键的发展，在均匀分布理论，这个项目的重点，利用应用程序的真实的分析，功能分析，特别是谐波分析。与此同时，历史上被其他领域的专家忽视的许多现代谐波分析方法，利用均匀分布理论，才开始在他们的领域站稳脚跟。对于均匀分布理论，特别是在更高的维度中，许多至关重要的问题仍然是开放的。该项目研究的问题包括高维最优差异的精确渐近性问题、一比特压缩感知、度量空间的嵌入、差异与能量最小化之间的相互作用、均匀分布点集的构造（低差异，体积，能量最小化，格子），差异估计和函数空间中的数值积分，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，包装）。在这些问题上的进展必须与分析和其他领域的重要问题和成就的进展紧密交织在一起，把一个明显不相关的问题和主题拼凑在一起。该项目的成果预计将影响数学的几个领域，用新的结果，思想和方法丰富和交叉它们。

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.