Extremal Point Configurations

The University Of Texas Rio Grande Valley, Edinburg TX

Investigators

Abstract

This project is devoted to the study of extremal point configurations, both in continuous cases such as Euclidean spaces or spheres and in discrete spaces. One particular area of focus is packing problems. A typical question in this area is to find the most efficient, or dense, packing. These types of questions have been well known in mathematics and science for a long time, starting with the Kepler conjecture about the densest sphere packing in three dimensions and with the kissing number problem that was the subject of disagreement between Isaac Newton and David Gregory at the end of the 17th century. A second focus area is the search for configurations that minimize energy. Probably the most famous example of this type is the Thomson problem asking to find the location of electrons in the sphere with the smallest cumulative electrostatic energy. The PI plans to mentor both undergraduate and graduate students and reach out to a wider audience by organizing research talks and public lectures. In more detail, this project will focus on the study and applications of extremal point configurations. One direction of research concerns sets with few pairwise distances. In the discrete case, sets with few distances are the subject of the Erdős–Ko–Rado and similar theorems. In the continuous case, one of the important objects of study is the set of equiangular lines. The PI will apply and extend the general method of finding upper bounds on sets with prescribed distances in two-point homogeneous spaces, including both the discrete and continuous regimes. A second direction of research concerns plank covering problems, that is, coverings of convex regions in Euclidean space by the regions between pairs of parallel hyperplanes. These questions go back to the Tarski problem and the Fejes Tóth zone conjecture. Recently, a version of the polynomial method brought several far-reaching generalizations of the results in this area. The PI will further develop this method and apply it to plank coverings and similar problems. Finally, a third direction of research is the study of energetically optimal configurations using a variety of analytic and optimization methods, including linear programming and semi-definite programming approach, with the goal of solving relevant problems in discrete and convex geometry. 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.