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.
