How error correcting codes linear programming methods and modular forms can help us to understand higher dimensional sphere packings
Abstract
In this project we are firstly concerned with an old and hard problem called The Sphere Packing Problem. The question is simple: What is the best way of stacking oranges in a supermarket? We all intuitively know the answer to this question in three dimensions. What is remarkable is that this problem was proposed by Kepler in 1611 and only solved in 1998 by Hales, in a famous and large computer-assisted proof. Later in 2016, the problem in dimensions 8 and 24 was solved by Viazovska, introducing new techniques from the theory of modular forms. Viazovska received the Fields medal (Nobel prize equivalent in mathematics) in 2022 for her accomplishments.Inthe present proposal we outline a new method for attacking this problem in higher dimensions. We propose a two- step strategy that combines exact analytic solutions, optimization methods and computer-assisted calculations to tackle the problem in dimensions 4, 5,6, 7 and 16, 32, 40, 48. The idea consists in finding first a stepping-stone for a secondary computer-assisted part, where we establish first that a certain configuration is optimal under additional geometric constraints, which we call the #second-best# configuration. We then show that the space of configurations better then the second-best is amenable to a finite-cases reduction and optimization methods, such as semidefinite-programming.We also propose a novel way of attacking the asymptotic problem of sphere packings, that is, the problem of estimating how fast the density of the best sphere packing in a given dimension N grows as N tends to infinity. This is in turn related with certain sign uncertainty principles and their asymptotic behavior. We propose to use this new connection to provide a new asymptotic upper bound for the sphere packing density of N -dimensional space.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Nov 09, 2024
- Source ID
- N629092412126
Entities
People
- Felipe Goncalves
Organizations
- Office of Naval Research
- United States Navy