Sphere packings and related extremal problems

这一建议致力于度量空间中的球填充问题、其他点分配问题以及相关的组合对象问题。通过在美国和国外的会议和座谈介绍调查结果，将扩大对这些主题的讨论，并促进这方面的进一步发展。同样，研究人员希望为教育目标做出贡献，告知下一代数学家，并激励他们进行自己的相关研究。研究人员将使用两种主要方法来分析极点分配。拥堵或不可约图的方法可以追溯到舒特和范德瓦尔登、费耶斯·托斯和丹泽，并与刚性理论直接相关。用正定约束的方法来分析点组态的性质，并求出点组态大小的界，依赖于勋伯格、博克纳和德尔沙特的经典工作。PI和co-PI的目的是解决n值大于13的Tammes问题，对二维球面上的不可约图进行分类，并描述其他曲面(包括n大于8的方形平坦环面)和更高维上的最优球面填充。研究人员还将尝试量化正定松弛对点集描述和已知Delsarte编码界限的准确性的影响。通过对少距离集的分析，PI和co-PI将为经典的组合问题提供新的工具，而研究这类集的几何思想可能会导致寻找强正则图和本原结合方案上的约束的新方法。Delaunay三角剖分为球面填充和覆盖的分析提供了一个重要的工具；PI和co-PI旨在将已知结果扩展到更大的点构型集，并寻找Delaunay三角剖分总是最优的新的全维泛函。

This proposal is devoted to the problems of sphere packings in metric spaces, other point allocations, and related combinatorial objects. The presentation of the findings through conferences and colloquia, both in the USA and abroad, will broaden the discussion of these topics and inspire further developments therein. Likewise the investigators expect to contribute to the educational goal of informing the next generation of mathematicians and inspiring them to conduct related research of their own.The investigators will employ two major methods of analyzing extremal point allocations. The method of jammed or irreducible graphs goes back to Schutte and van der Waerden, Fejes Toth, and Danzer, and is directly related to the rigidity theory. The method of positive definite constraints that has been used to analyze the properties of point configurations, and to derive bounds on their size, relies on classical works of Schoenberg, Bochner, and Delsarte. The PI and co-PI intend to solve the Tammes problem for values of n larger than 13, classify irreducible graphs on a two-dimensional sphere, and describe optimal sphere packings on other surfaces (including a square flat torus with n larger than 8) and in higher dimensions. The investigators will also attempt to quantify the impact of the positive definite relaxation on the description of point sets and the accuracy of the known Delsarte bounds on codes. Through the proposed analysis of few-distance sets, the PI and co-PI will contribute new tools to a classic combinatorial problem; while geometric ideas used in the study of such sets could lead to new approaches to finding constraints on strongly regular graphs and primitive association schemes. Delaunay triangulations provide an important tool for analysis of sphere packings and coverings; the PI and co-PI intend to extend known results to a larger set of point configurations and to find new all-dimensional functionals for which a Delaunay triangulation is always optimal.