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

The University Of Texas Rio Grande Valley, Edinburg TX

Investigators

Abstract

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.