Explore the fascinating world of sphere packing problems in this comprehensive lecture by Abhinav Kumar at the International Centre for Theoretical Sciences. Delve into the centuries-old geometric challenge of arranging spheres in space to maximize density, and discover its connections to number theory, communication theory, and physics. Learn about known solutions in low dimensions, including the hexagonal planar packing and the Hales theorem. Investigate higher-dimensional packings, non-lattice arrangements, and the intriguing E8 and Leech lattices. Examine upper and lower bounds, lattice theory, and asymptotic behavior in high dimensions. Explore related concepts like spherical codes, the kissing number problem, and energy minimization. Gain insights into open problems and recent advancements in this active area of mathematical research.
Introduction to Sphere Packing Problems by Abhinav Kumar
International Centre for Theoretical Sciences via YouTube
Overview
Syllabus
Introduction Sphere Packing problem
What is this your packing problem?
Collection of spheres
What is the density of these spheres?
Some answers which are known only in some small dimensions
Stack layers of the hexagonal lattice
Hexagonal planar packing
Why is this the best?
These are the densest proof by hales & collaborators
What we can guess in some higher dimensions?
Low dimensional lattices
What are all the densest packing's in low dimension and fiber in construction?
Modular lattices
Infinity many possibilities and they are obtained by suitable coloring of a packing
For along time it was suspected that E8 lattice packing was the unique densest
All kinds of strange things happen
First dimension the densest known packing is a non lattice & its consists of 40 trsanslates
What about higher dimensions?
What happens between ten and twenty four?
Upper and easy lower bound
Lower bound
Lattices
What is the densest lattice in dimensions?
Flow conjecture in high dimensions
Asymptotics
What the best lattices should look like in very high dimensions?
Spherical codes
Kissing number problem
Energy Minimization
Energy minimizer's obviously depend on f
Open problem
Taught by
International Centre for Theoretical Sciences
