Lattices, sphere packings and spherical codes

121 Pages·2012·0.75 MB·English

by  Abhinav Kumar

Lattices, sphere packings and spherical codes: geometric
optimization problems
Abhinav Kumar
MIT
November 25, 2012
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 1/46
Sphere packings
Definition
A sphere packing in Rn is a collection of spheres/balls of equal size which
do not overlap (except for touching). The density of a sphere packing is
the volume fraction of space occupied by the balls.
The main question is to find a/the densest packing in Rn.
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 2/46
Good sphere packings
In dimension 1, we can achieve density 1 by laying intervals end to end.
In dimension 2, the best possible is by using the hexagonal lattice. [Fejes
To´th, 1940]
∼
✎☞✎☞✎✎☞☞✎☞
✍✎✌✍☞✎✌✍✎☞✍✌☞✎✌✍☞✌
✎✍☞✎✌✍☞✎✌✍✎☞✌✍☞✎✌☞
✍✎✌✍☞✎✌✍✎☞✍✌☞✎✌✍☞✌
✎✍☞✎✌✍☞✎✌✍✎☞✌✍☞✎✌☞
✍✌✍✌✍✍✌✌✍✌
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 3/46
Good sphere packings
In dimension 1, we can achieve density 1 by laying intervals end to end.
In dimension 2, the best possible is by using the hexagonal lattice. [Fejes
To´th, 1940]
∼
✎☞✎☞✎✎☞☞✎☞
✍✎✌✍☞✎✌✍✎☞✍✌☞✎✌✍☞✌
✎✍☞✎✌✍☞✎✌✍✎☞✌✍☞✎✌☞
✍✎✌✍☞✎✌✍✎☞✍✌☞✎✌✍☞✌
✎✍☞✎✌✍☞✎✌✍✎☞✌✍☞✎✌☞
✍✌✍✌✍✍✌✌✍✌
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 3/46
Good sphere packings II
In dimension 3, the best possible way is to stack layers of the solution in 2
dimensions. [Hales, 1998]
∼
✎☞✎☞✎✎☞☞✎☞
✍✎✌✍☞✎✌✍✎☞✍✌☞✎✌✍☞✌
✎☞✎✎☞☞✎☞
✎✍☞✎♠✌✍☞✎♠✌✍✎☞♠✌✍☞✎♠✌☞
✍✍✎✌✎✍✌☞✍✎♠☞✌✎✍✌✍✎☞♠✎☞✍✌✌✍☞✎♠☞✎✌✍✌☞♠☞✌
✎✍☞✍✎✌✍✌☞✍✎✌✍✍✌✎☞✌✍✌✍☞✎✌✌☞
✍✌✍✌✍✍✌✌✍✌
There are infinitely (in fact, uncountably) many ways of doing this, the
Barlow packings.
In dimensions 4, we have some guesses for the densest sphere packing.
≥
But we can’t prove them. In low dimensions, the best known sphere
packings come from lattices.
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 4/46
Good sphere packings II
In dimension 3, the best possible way is to stack layers of the solution in 2
dimensions. [Hales, 1998]
∼
✎☞✎☞✎✎☞☞✎☞
✍✎✌✍☞✎✌✍✎☞✍✌☞✎✌✍☞✌
✎☞✎✎☞☞✎☞
✎✍☞✎♠✌✍☞✎♠✌✍✎☞♠✌✍☞✎♠✌☞
✍✍✎✌✎✍✌☞✍✎♠☞✌✎✍✌✍✎☞♠✎☞✍✌✌✍☞✎♠☞✎✌✍✌☞♠☞✌
✎✍☞✍✎✌✍✌☞✍✎✌✍✍✌✎☞✌✍✌✍☞✎✌✌☞
✍✌✍✌✍✍✌✌✍✌
There are infinitely (in fact, uncountably) many ways of doing this, the
Barlow packings.
In dimensions 4, we have some guesses for the densest sphere packing.
≥
But we can’t prove them. In low dimensions, the best known sphere
packings come from lattices.
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 4/46
Lattices
Definition
A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n
linearly independent vectors of Rn.
Example
Zn Rn
⊂
Root lattices A ,D ,E .
n n n
Lattices are related to quadratic forms: if b ,...,b is a basis of the
1 n
lattice Λ Rn, then Q(x ,...,x ) = x b + +x b 2 is a positive
1 n 1 1 n n
⊂ || ··· ||
definite quadratic form.
Lattices up to isometry Quadratic forms mod change of coords
{ } ↔ { }
On(R) GLn(R)/GLn(Z) ∼= Sym2(Rn)+/SLn(Z).
\
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 5/46
Lattices
Definition
A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n
linearly independent vectors of Rn.
Example
Zn Rn
⊂
Root lattices A ,D ,E .
n n n
Lattices are related to quadratic forms: if b ,...,b is a basis of the
1 n
lattice Λ Rn, then Q(x ,...,x ) = x b + +x b 2 is a positive
1 n 1 1 n n
⊂ || ··· ||
definite quadratic form.
Lattices up to isometry Quadratic forms mod change of coords
{ } ↔ { }
On(R) GLn(R)/GLn(Z) ∼= Sym2(Rn)+/SLn(Z).
\
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 5/46
Lattices
Definition
A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n
linearly independent vectors of Rn.
Example
Zn Rn
⊂
Root lattices A ,D ,E .
n n n
Lattices are related to quadratic forms: if b ,...,b is a basis of the
1 n
lattice Λ Rn, then Q(x ,...,x ) = x b + +x b 2 is a positive
1 n 1 1 n n
⊂ || ··· ||
definite quadratic form.
Lattices up to isometry Quadratic forms mod change of coords
{ } ↔ { }
On(R) GLn(R)/GLn(Z) ∼= Sym2(Rn)+/SLn(Z).
\
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 5/46
Lattices
Definition
A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n
linearly independent vectors of Rn.
Example
Zn Rn
⊂
Root lattices A ,D ,E .
n n n
Lattices are related to quadratic forms: if b ,...,b is a basis of the
1 n
lattice Λ Rn, then Q(x ,...,x ) = x b + +x b 2 is a positive
1 n 1 1 n n
⊂ || ··· ||
definite quadratic form.
Lattices up to isometry Quadratic forms mod change of coords
{ } ↔ { }
On(R) GLn(R)/GLn(Z) ∼= Sym2(Rn)+/SLn(Z).
\
AbhinavKumar (MIT) Geometricoptimizationproblems November25,2012 5/46