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

RUI：欧几里得空间、球体和离散空间中的点配置

• 批准号: 2054536
• 负责人: Alexey Glazyrin
• 金额: $ 18万
• 依托单位: The University of Texas Rio Grande Valley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054536&HistoricalAwards=false
• 关键词: RUI; Point; Configurations; Euclidean; Spaces

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.

本项目主要研究欧氏空间和球面中的极值离散点构形问题。这些问题自经典的开普勒猜想和接吻数问题以来一直存在，这两个问题都起源于世纪。关于三维空间中最致密球体堆积的开普勒猜想可以追溯到沃尔特·罗利，他要求确定在船只甲板上堆叠炮弹的最佳方式。接吻数问题是艾萨克·牛顿和大卫·格雷戈里之间一次著名讨论的主题。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课题，在数学优化、近似理论、编码理论、信息论、材料科学和晶体学等许多领域都有应用。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，研究人员还计划通过与德克萨斯大学格兰德河谷的STEM教育卓越中心的合作，接触到广泛的本科生。该中心的目标是加强STEM学术项目并增加STEM毕业生的数量，特别是来自代表性不足群体的毕业生的数量。该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，主要的方法依赖于这样一个事实，即在某些条件下，最佳点配置通过线性或半定条件受到空间对称性的约束。主要研究者建立的两点齐性空间中的少距离集上界的方法，将为解决经典组合问题提供新的工具。预计这种方法的推广版本可能会导致新的边界球包装，并可以适用于许多不同的情况下。对于另一组问题，组合和数论对象（图，格等）的对称性。隐含着相应点集几何最优性。本项目建议的方法是使用分析方法和未知配置的假设最优性来构建它们或证明它们的存在/不存在。PI还将使用软包装获得新的范围为各种包装和覆盖问题和调查的一般问题，找到最大密度的软包装在各种settings.This奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

项目成果

Covering by Planks and Avoiding Zeros of Polynomials

用木板覆盖并避免多项式的零点

• DOI: 10.1093/imrn/rnac259
• 发表时间: 2022
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Glazyrin, Alexey;Karasev, Roman;Polyanskii, Alexandr
• 通讯作者: Polyanskii, Alexandr

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

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

Covering by homothets and illuminating convex bodies

由同位覆盖和照明凸体

• DOI: 10.1090/proc/15516
• 发表时间: 2022
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Glazyrin, Alexey