CAREER: Lattices and Sphere Packings, Arithmetic Geometry and Computational Number Theory

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

In this project, the PI, Kumar, will investigate the topics of sphere packings, arithmetic geometry and computational number theory, and the various areas of overlap of these directions of research. The PI and his collaborators have worked on the problem of finding the densest packings in 8 and 24 dimensions, using the technique of linear programming bounds. They have studied the sphere packing and associated coding problems in context of potential energy minimization. This has led to various new questions and techniques, such as the concept of universal optimality, gradient descent in the space of packings, and inverse problems which may have applications to molecular self-assembly. This project will explore some of the questions that arose in previous investigations, as well as other fundamental questions such as improvement of asymptotic bounds on the density of sphere packings. Another component of the project is arithmetic geometry, especially the study of K3 surfaces and their automorphisms, such as Shioda-Inose structures. The PI also proposes to study arithmetic applications such as the description of modular curves and surfaces, Galois representations and the computation of modular forms and elliptic curves of high rank. The project also encompasses related questions in computational number theory. Lattices are a unifying theme of the topics of the proposal, and one of the broad goals of the project is to understand explicitly families of interesting lattices in high dimensions. The "greengrocer's problem" of packing equal sized non-overlapping spheres efficiently in space is a classical problem in geometry. Nevertheless, it turns out to have surprising and beautiful connections with various parts of mathematics, physics, and computer science. Its theoretical and practical ramifications extend to Lie algebras, exceptional finite groups, quadratic forms, coding theory, cryptography and energy minimization in physics. This project will seek to further our understanding of sphere packings in high dimensions as well as explore and exploit these connections. Arithmetic geometry seeks to understand the properties of the natural numbers using the powerful tools of algebraic geometry. One of the most spectacular successes of twentieth century algebraic geometry is Faltings' Theorem (erstwhile Mordell's Conjecture), which provides the final link in a trichotomy which links the number of rational points on an algebraic curve to its topological genus, and roughly establishes how many solutions we can expect to an equation linking two variables. The analogous question of how many solutions we can expect to an equation linking three or more variables is still quite a mystery. Some of the questions the PI intends to investigate involve the arithmetic of surfaces, especially K3 surfaces, which are important in mathematics and physics. The project also hopes to make progress on the still murkier computational aspects, such as how to find these solutions. The PI will also develop courses and seminars in topics related to the proposal, and aim to involve graduate and undergraduate students actively in this research. Other aims of the project will be to develop algorithms and open-source code, as well as a catalog of interesting codes in different spaces.