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Introduction Sphere Packing problem
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Introduction to Sphere Packing Problems by Abhinav Kumar
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- 1 Introduction Sphere Packing problem
- 2 What is this your packing problem?
- 3 Collection of spheres
- 4 What is the density of these spheres?
- 5 Some answers which are known only in some small dimensions
- 6 Stack layers of the hexagonal lattice
- 7 Hexagonal planar packing
- 8 Why is this the best?
- 9 These are the densest proof by hales & collaborators
- 10 What we can guess in some higher dimensions?
- 11 Low dimensional lattices
- 12 What are all the densest packing's in low dimension and fiber in construction?
- 13 Modular lattices
- 14 Infinity many possibilities and they are obtained by suitable coloring of a packing
- 15 For along time it was suspected that E8 lattice packing was the unique densest
- 16 All kinds of strange things happen
- 17 First dimension the densest known packing is a non lattice & its consists of 40 trsanslates
- 18 What about higher dimensions?
- 19 What happens between ten and twenty four?
- 20 Upper and easy lower bound
- 21 Lower bound
- 22 Lattices
- 23 What is the densest lattice in dimensions?
- 24 Flow conjecture in high dimensions
- 25 Asymptotics
- 26 What the best lattices should look like in very high dimensions?
- 27 Spherical codes
- 28 Kissing number problem
- 29 Energy Minimization
- 30 Energy minimizer's obviously depend on f
- 31 Open problem
