Energy minimization for lattices and periodic configurations

Energy minimization for lattices and periodic configurations, and formal duality
Abhinav Kumar
MIT
November 14, 2011

joint work with Henry Cohn and Achill Schu¨rmann

Abhinav Kumar (MIT)

Potential

November 14, 2011 1 / 20

Sphere packings

Sphere packing problem: What is (a/the) densest sphere packing in n dimensions?
In low dimensions, the best densities known are achieved by lattice packings.

n Λ due to

123 A1 A2 A3
Gauss

45
D4 D5 KorkineZolotareff

678 E6 E7 E8
Blichfeldt

24 Leech Cohn-K.

Abhinav Kumar (MIT)

Potential

November 14, 2011 2 / 20

Sphere packings

Sphere packing problem: What is (a/the) densest sphere packing in n dimensions?
In low dimensions, the best densities known are achieved by lattice packings.

n Λ due to

123 A1 A2 A3
Gauss

45
D4 D5 KorkineZolotareff

678 E6 E7 E8
Blichfeldt

24 Leech Cohn-K.

Abhinav Kumar (MIT)

Potential

November 14, 2011 2 / 20

Low dimensions
n = 1: lay intervals end to end (density 1). n = 2: hexagonal or A2 arrangement [Fejes-T´oth 1940]
✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
This is the unique densest periodic packing.

Abhinav Kumar (MIT)

Potential

November 14, 2011 3 / 20

Low dimensions
n = 1: lay intervals end to end (density 1). n = 2: hexagonal or A2 arrangement [Fejes-T´oth 1940]
✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✍☞ ✎✌ ✍☞ ✎✌ ✍✎ ☞✌ ✍☞ ✎✌☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
This is the unique densest periodic packing.

Abhinav Kumar (MIT)

Potential

November 14, 2011 3 / 20

Barlow packings
n = 3 : stack layers of the solution in 2 dimensions. [Hales 1998] ✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✎ ✍☞ ✎ ♠☞ ✌ ✎ ✍☞ ✎ ♠✌ ✎ ✍ ☞✎ ☞ ♠☞ ✎ ✌ ✍☞ ✎ ♠☞ ✌☞ ✍✍ ✎✌ ✎ ✍✌ ☞ ✍ ✎ ♠☞ ✌ ✎ ✍✌ ✍ ✎ ☞ ♠✎ ☞ ✍ ✌✌ ✍ ☞ ✎ ♠☞ ✎ ✌ ✍✌ ☞ ♠☞ ✌ ✎✍☞ ✍ ✎✌ ✍✌ ☞ ✍ ✎✌ ✍✍ ✌ ✎ ☞✌ ✍✌ ✍ ☞ ✎✌✌ ☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
Uncountably many ways of doing this, the Barlow packings. Even in dimensions 5, 6, 7, densest lattices have (uncountably many) competitors.

Abhinav Kumar (MIT)

Potential

November 14, 2011 4 / 20

Barlow packings
n = 3 : stack layers of the solution in 2 dimensions. [Hales 1998] ✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✎ ✍☞ ✎ ♠☞ ✌ ✎ ✍☞ ✎ ♠✌ ✎ ✍ ☞✎ ☞ ♠☞ ✎ ✌ ✍☞ ✎ ♠☞ ✌☞ ✍✍ ✎✌ ✎ ✍✌ ☞ ✍ ✎ ♠☞ ✌ ✎ ✍✌ ✍ ✎ ☞ ♠✎ ☞ ✍ ✌✌ ✍ ☞ ✎ ♠☞ ✎ ✌ ✍✌ ☞ ♠☞ ✌ ✎✍☞ ✍ ✎✌ ✍✌ ☞ ✍ ✎✌ ✍✍ ✌ ✎ ☞✌ ✍✌ ✍ ☞ ✎✌✌ ☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
Uncountably many ways of doing this, the Barlow packings. Even in dimensions 5, 6, 7, densest lattices have (uncountably many) competitors.

Abhinav Kumar (MIT)

Potential

November 14, 2011 4 / 20

Barlow packings
n = 3 : stack layers of the solution in 2 dimensions. [Hales 1998] ✎☞ ✎☞ ✎✎ ☞☞ ✎☞ ✍✎✌ ✍☞ ✎✌ ✍✎ ☞✍ ✌☞ ✎✌ ✍☞✌ ✎✎ ✍☞ ✎ ♠☞ ✌ ✎ ✍☞ ✎ ♠✌ ✎ ✍ ☞✎ ☞ ♠☞ ✎ ✌ ✍☞ ✎ ♠☞ ✌☞ ✍✍ ✎✌ ✎ ✍✌ ☞ ✍ ✎ ♠☞ ✌ ✎ ✍✌ ✍ ✎ ☞ ♠✎ ☞ ✍ ✌✌ ✍ ☞ ✎ ♠☞ ✎ ✌ ✍✌ ☞ ♠☞ ✌ ✎✍☞ ✍ ✎✌ ✍✌ ☞ ✍ ✎✌ ✍✍ ✌ ✎ ☞✌ ✍✌ ✍ ☞ ✎✌✌ ☞ ✍✌ ✍✌ ✍✍ ✌✌ ✍✌
Uncountably many ways of doing this, the Barlow packings. Even in dimensions 5, 6, 7, densest lattices have (uncountably many) competitors.

Abhinav Kumar (MIT)

Potential

November 14, 2011 4 / 20

Root lattices
An (simplex lattice) = {x ∈ Zn+1 | xi = 0}, inside the zero-sum hyperplane {x ∈ Rn+1 | xi = 0} ∼= Rn. Dn (checkerboard lattice) = {x ∈ Zn | xi ≡ 0 (mod 2)} E8 = D8 (D8 + (1/2, . . . , 1/2)). E7 = orthogonal complement of A1 inside E8. E6 = orthogonal complement of A2 inside E8.

Abhinav Kumar (MIT)

Potential

November 14, 2011 5 / 20

High dimensions
In higher dimensions, we believe the densest sphere packings don’t come from lattices. Example In R10 the densest known is the Best packing, 40 translates of a lattice.
But do believe the densest packings can be achieved by periodic packings (Zassenhaus conjecture). Can provably come arbitrarily close for packing density. Trivial Minkowski bound implies ∃ packing with density ≥ 1/2n, but no explicit constructions known.

Abhinav Kumar (MIT)

Potential

November 14, 2011 6 / 20