Configurations of Extremal Even Unimodular Lattices Scott D. - PowerPoint PPT Presentation

Configurations of Extremal Even Unimodular Lattices Scott D. Kominers Harvard Mathematics Department Brown University SUMS March 8, 2008 Configurations of ExtremalEven Unimodular Lattices – p. 1/33

Key Terms Configurations of ExtremalEven Unimodular Lattices – p. 2/33

Key Terms Lattice L Configurations of ExtremalEven Unimodular Lattices – p. 2/33

Key Terms Lattice L — “integer vector space” Configurations of ExtremalEven Unimodular Lattices – p. 2/33

Key Terms Lattice L — “integer vector space” free Z -module Configurations of ExtremalEven Unimodular Lattices – p. 2/33

Key Terms Lattice L — “integer vector space” free Z -module with an inner product �· , ·� : L × L → R Configurations of ExtremalEven Unimodular Lattices – p. 2/33

Key Terms Lattice L — “integer vector space” Configurations of ExtremalEven Unimodular Lattices – p. 3/33

Key Terms Lattice L — “integer vector space” rank Configurations of ExtremalEven Unimodular Lattices – p. 3/33

Key Terms Lattice L — “integer vector space” rank number of basis vectors Configurations of ExtremalEven Unimodular Lattices – p. 3/33

Key Terms Lattice L — “integer vector space” even Configurations of ExtremalEven Unimodular Lattices – p. 4/33

Key Terms Lattice L — “integer vector space” even � x, x � ∈ 2 Z for all x ∈ L Configurations of ExtremalEven Unimodular Lattices – p. 4/33

Key Terms Lattice L — “integer vector space” even � x, x � ∈ 2 Z for all x ∈ L all vectors have even squared length Configurations of ExtremalEven Unimodular Lattices – p. 4/33

Key Terms Lattice L — “integer vector space” unimodular Configurations of ExtremalEven Unimodular Lattices – p. 5/33

Key Terms Lattice L — “integer vector space” unimodular L is “self-dual” Configurations of ExtremalEven Unimodular Lattices – p. 5/33

Key Terms Lattice L — “integer vector space” unimodular L is “self-dual” L ’s matrix has determinant 1 Configurations of ExtremalEven Unimodular Lattices – p. 5/33

Key Terms Review: even unimodular lattice L Configurations of ExtremalEven Unimodular Lattices – p. 6/33

Key Terms Review: even unimodular lattice L all vectors have even squared length Configurations of ExtremalEven Unimodular Lattices – p. 7/33

Key Terms Review: even unimodular lattice L “self-dual” Configurations of ExtremalEven Unimodular Lattices – p. 8/33

Key Terms Review: even unimodular lattice L “integer vector space” Configurations of ExtremalEven Unimodular Lattices – p. 9/33

Key Terms even unimodular lattice L ... Configurations of ExtremalEven Unimodular Lattices – p. 10/33

Key Terms even unimodular lattice L ... “rare” Configurations of ExtremalEven Unimodular Lattices – p. 10/33

Key Terms even unimodular lattice L ... “rare” rank( L ) = 8 n Configurations of ExtremalEven Unimodular Lattices – p. 10/33

Key Terms even unimodular lattice L ... Configurations of ExtremalEven Unimodular Lattices – p. 11/33

Key Terms even unimodular lattice L ... extremal Configurations of ExtremalEven Unimodular Lattices – p. 11/33

Key Terms even unimodular lattice L ... extremal shortest vector Configurations of ExtremalEven Unimodular Lattices – p. 11/33

Key Terms even unimodular lattice L ... extremal shortest vector ↔ as long as possible Configurations of ExtremalEven Unimodular Lattices – p. 11/33

We care about... extremal even unimodular lattices Configurations of ExtremalEven Unimodular Lattices – p. 12/33

We care because... extremal even unimodular lattices Configurations of ExtremalEven Unimodular Lattices – p. 13/33

We care because... extremal even unimodular lattices are useful Configurations of ExtremalEven Unimodular Lattices – p. 13/33

We care because... extremal even unimodular lattices are useful Sphere-packing problems Configurations of ExtremalEven Unimodular Lattices – p. 13/33

We care because... extremal even unimodular lattices are useful Sphere-packing problems Storing n -dimensional oranges Configurations of ExtremalEven Unimodular Lattices – p. 13/33

We care because... extremal even unimodular lattices are useful Sphere-packing problems Storing n -dimensional oranges Chemical lattices (in dimensions n ≤ 3 ) Configurations of ExtremalEven Unimodular Lattices – p. 13/33

We care because... extremal even unimodular lattices are useful Sphere-packing problems Storing n -dimensional oranges Chemical lattices (in dimensions n ≤ 3 ) (Continuous) communication decoding Configurations of ExtremalEven Unimodular Lattices – p. 13/33

Methods: Theta Functions Configurations of ExtremalEven Unimodular Lattices – p. 14/33

Methods: Theta Functions Slogan: “The theta function of a lattice L encodes the lengths of L ’s vectors.” Configurations of ExtremalEven Unimodular Lattices – p. 14/33

Methods: Theta Functions Slogan: “The theta function of a lattice L encodes the lengths of L ’s vectors.” How: “dot product � x, x � ” ⇐ ⇒ “length” Configurations of ExtremalEven Unimodular Lattices – p. 14/33

Methods: Theta Functions Slogan: “The theta function of a lattice L encodes the lengths of L ’s vectors.” How: “dot product � x, x � ” ⇐ ⇒ “length” What: ∞ e iπτ � x,x � = � � a k e iπτ ( k ) Θ L ( τ ) = x ∈ L k =1 Configurations of ExtremalEven Unimodular Lattices – p. 14/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Configurations of ExtremalEven Unimodular Lattices – p. 15/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Example: Θ Z 2 ( τ ) Configurations of ExtremalEven Unimodular Lattices – p. 15/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Example: e iπτ � x,x � = 1 + 4 e iπτ + 4 e 2 iπτ + 4 e 4 iπτ + · · · � Θ Z 2 ( τ ) = x ∈ L Configurations of ExtremalEven Unimodular Lattices – p. 15/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Why we care: Theta functions of even unimodular lattices are examples of modular forms . Configurations of ExtremalEven Unimodular Lattices – p. 16/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Why we care: Theta functions of even unimodular lattices are examples of modular forms . We can write down the spaces of modular forms explicitly. Configurations of ExtremalEven Unimodular Lattices – p. 16/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Why we care: We can write down the spaces of modular forms explicitly. Configurations of ExtremalEven Unimodular Lattices – p. 17/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Why we care: We can write down the spaces of modular forms explicitly. We can therefore study the function Θ L ... Configurations of ExtremalEven Unimodular Lattices – p. 17/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � encodes the lengths of L ’s vectors.” Why we care: We can write down the spaces of modular forms explicitly. We can therefore study the function Θ L ... ...even if we cannot write down a basis for L . Configurations of ExtremalEven Unimodular Lattices – p. 17/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � is a modular form which encodes the lengths of L ’s vectors.” Configurations of ExtremalEven Unimodular Lattices – p. 18/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � is a modular form which encodes the lengths of L ’s vectors.” What we do: We study weighted theta functions � x ∈ L P ( x ) e iπτ � x,x � (also modular forms) and obtain linear systems of equations... Configurations of ExtremalEven Unimodular Lattices – p. 18/33

Methods: Theta Functions Slogan: x ∈ L e iπτ � x,x � of a lattice L “The theta function � is a modular form which encodes the lengths of L ’s vectors.” What we do: We study weighted theta functions � x ∈ L P ( x ) e iπτ � x,x � (also modular forms) and obtain linear systems of equations... ...which give combinatorial information about lattice vectors. Configurations of ExtremalEven Unimodular Lattices – p. 18/33

Results Configurations of ExtremalEven Unimodular Lattices – p. 19/33

Results Theorem. Configurations of ExtremalEven Unimodular Lattices – p. 20/33

Results Theorem Template. Configurations of ExtremalEven Unimodular Lattices – p. 21/33