Extremal Problems in Discrete Spherical Geometry

Arizona State University, Scottsdale AZ

Investigators

Abstract

Spheres, familiar shapes encountered in everyday life, from sports balls to celestial bodies, conceal deep and complex mathematical questions within their seemingly simple structures. This project explores challenging problems in discrete geometry specifically related to spheres. Understanding these problems not only advances mathematical knowledge but also has broad implications across science and technology, influencing fields such as data science, communications, and materials science. Central to this project is the integration of advanced mathematical research into education. By developing innovative curricula that blend theoretical exploration with practical problem-solving, this initiative will prepare undergraduate and high school students with critical thinking and analytical skills, fostering the next generation of mathematicians and scientists. The mathematical focus of this project addresses fundamental and intricate problems within spherical discrete geometry, including equiangular lines, spherical two-distance sets, plank coverings of spheres, the dual of the Komlós conjecture, and rainbow problems in combinatorics. These problems, while easily stated, require innovative approaches that go beyond traditional methods. The research draws on techniques from combinatorics, optimization, and harmonic analysis, and actively fosters collaboration across mathematics, physics, and computer science. Interactions with physics provide insight into geometric configurations that model physical systems, while connections with computer science inform algorithmic and computational aspects related to coding theory and data representation. Additionally, the project emphasizes the dissemination of mathematics through workshops and conferences, contributing valuable tools and perspectives to the broader mathematical community. 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.