ICTS

Sphere packing is a centuries-old problem in geometry, with many connections to other branches of mathematics (number theory, communication theory, Lie theory, optimization) as well as, physics, material science and chemistry. In the recent years, there have been some remarkable advances in higher-dimensional sphere packing and the associated physics question of energy minimization, coming from methods of modular forms and optimization. Sphere packing continues to be an active area of research, with a long list of interesting and accessible questions.  The goal of this meeting is to introduce the subject to a diverse group of mathematicians, and to updated them on the new developments and research problems faced in this area of research.

A series of lectures will address the different facets of sphere packing and questions associated with it. It will also highlight, new techniques and open-up the challenges and problems that one can come across. A total of ten lectures will be held over a period of five days and will cover the following topics;

1. Introduction to sphere packing and related topics - lattices, spherical codes, covering, energy minimization and theta functions.
2. Lower bounds - nice packings, codes and symmetry groups.
3. Quadratic forms and Hermite constant, reduction theory, LLL, Voronoi's theorem including perfect and eutectic lattices.
4. Upper bounds - Rogers, Kabatiansky-Levenshein, Cohn-Elkies LP bounds and asymptotics.
5. Harmonic analysis - positive definite kernels on spheres and Euclidean spaces.
6. Semidefinite programming bounds for codes and sphere packings with examples.
7. Energy minimization - setup, LP bounds, numerical results in low dimensions for spheres and Euclidean spaces, and universal optimality.
8. Modular and quasi modular forms, extremal lattices, and Siegel modular forms.
9. Proof of optimality of E8 and Leech sphere packings and sketch of proof of universal optimality.
10. Inverse problem - finding a potential function which is minimized by target discrete structures on sphere or space.

The lecturers will primarily be by Prof. Abhinav Kumar (Infosys Visiting Professor at ICTS-TIFR) with additional lectures by Prof. Tathagata Basak, Prof. Radhika Ganapathy, Prof. Mahesh Kakde, and Prof. E K Narayanan. 

Participation is by Invitation only.