RUI: Point Configurations in Euclidean Spaces, Spheres, and Discrete Spaces

本课题主要研究欧几里得空间和球面上的极值离散点构型问题。自经典的开普勒猜想和接吻数问题（这两个问题都起源于17世纪）以来，这些问题一直存在。开普勒关于三维空间中密度最大的球体的猜想可以追溯到沃尔特·罗利（Walter Raleigh），他要求确定在他的船甲板上堆放炮弹的最佳方式。接吻数问题是艾萨克·牛顿和大卫·格里高利之间著名讨论的主题。这些问题后来导致了组合学和其他领域的各种主题。目前，点构型是一个跨学科的研究课题，在数学优化、近似理论、编码理论、信息论、材料科学、晶体学等领域都有广泛的应用。该项目的目标是研究在特定条件下的最佳配置。通过这个项目，研究者还计划通过与德克萨斯大学格兰德谷分校卓越STEM教育中心的合作，接触到广泛的本科生受众。该中心的目标是加强STEM学术项目，增加STEM毕业生的数量，特别是那些来自代表性不足群体的毕业生。项目中考虑的所有主题和问题的统一主题是点集的最优性。对于一组问题，主要的方法依赖于在某些条件下，最优点配置是由线性或半定条件下的空间对称性约束的事实。由主要研究者建立的在两点齐次空间中寻找小距离集合上界的方法将为解决经典组合问题提供新的工具。我们期望这种方法的广义版本可能会导致球体填料的新界限，并可适用于许多不同的情况。对于另一组问题，组合和数论对象（图，格等）的对称性意味着相应点集的某些几何最优性。本项目建议的方法是使用解析方法和未知配置的假设最优性来构造它们或证明它们的存在/不存在。PI还将使用软填料来获得各种填料和覆盖问题的新边界，并研究在各种设置下寻找软填料最大密度的一般问题。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。

This project focuses on the problems of extremal discrete point configurations in Euclidean space and spheres. These have persisted since the classical Kepler conjecture and the kissing number problem, both of which originated in the 17th century. The Kepler conjecture on densest sphere packings in three dimensions goes back to Walter Raleigh who asked to determine the best way to stack cannonballs on the decks of his ships. The kissing number problem was the subject of a famous discussion between Isaac Newton and David Gregory. Such questions later led to a variety of topics in combinatorics and other areas. Nowadays, point configurations is an interdisciplinary topic with applications in many areas such as mathematical optimization, approximation theory, coding theory, information theory, materials science, and crystallography. The goal of the project is to study configurations that are optimal under certain conditions. By this project, the investigator also plans to reach a wide audience of undergraduate students via the collaboration with the Center of Excellence in STEM Education of the University of Texas Rio Grande Valley. The goals of the Center are focused on strengthening STEM academic programs and increasing the number of STEM graduates, particularly those from underrepresented groups.The unifying theme for all the topics and problems considered in the project is the optimality of point sets. For one set of questions, the main approach relies on the fact that under some conditions optimal point configurations are constrained by space symmetries via linear or semidefinite conditions. The method of finding upper bounds on few-distance sets in two-point homogeneous spaces, established by the principal investigator, will provide new tools to address classical combinatorial problems. It is expected that the generalized version of this approach may lead to new bounds in sphere packings and can be applicable in many different contexts. For the other set of questions, symmetries of combinatorial and number-theoretic objects (graphs, lattices, etc.) imply certain geometric optimality of corresponding point sets. The approach suggested for this project is to use analytic methods and the hypothetical optimality of unknown configurations to construct them or prove their existence/non-existence. The PI will also use soft packings to obtain new bounds for a variety of packing and covering problems and investigate the general problem of finding maximal densities of soft packings in various settings.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.