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Recent breakthroughs in sphere packing Abhinav Kumar Stony Brook, ICTS November 8, 2019 Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 1 / 47 Sphere packings Definition A sphere packing in R n is a

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Recent breakthroughs in sphere packing

Abhinav Kumar

Stony Brook, ICTS

November 8, 2019

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 1 / 47

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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.

⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 2 / 47

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Sphere packing problem

Problem: Find a/the densest sphere packing(s) in Rn.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 3 / 47

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Sphere packing problem

Problem: Find a/the densest sphere packing(s) in Rn. In dimension 1, we can achieve density 1 by laying intervals end to end.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 3 / 47

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Sphere packing problem

Problem: Find a/the densest sphere packing(s) in Rn. 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 T´

  • th 1940]

✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 3 / 47

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Sphere packing problem II

In dimension 3, the best possible way is to stack layers of the solution in 2

  • dimensions. This is Kepler’s conjecture, now a theorem of Hales.

✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ✍✌ ✎☞ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠

There are infinitely (in fact, uncountably) many ways of doing this! These are the Barlow packings.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 4 / 47

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Face centered cubic packing

Image: Greg A L (Wikipedia), CC BY-SA 3.0 license Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 5 / 47

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Higher dimensions

In some higher dimensions, we have guesses for the densest sphere packings. Most of them arise from lattices.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 6 / 47

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Higher dimensions

In some higher dimensions, we have guesses for the densest sphere packings. Most of them arise from lattices. But (until very recently!) no proofs.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 6 / 47

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Higher dimensions

In some higher dimensions, we have guesses for the densest sphere packings. Most of them arise from lattices. But (until very recently!) no proofs. In very high dimensions (say ≥ 1000) densest packings are likely to be close to disordered.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 6 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples:

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples: Integer lattice Zn.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples: Integer lattice Zn. Checkerboard lattice Dn = {x ∈ Zn : xi even }

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples: Integer lattice Zn. Checkerboard lattice Dn = {x ∈ Zn : xi even } Simplex lattice An = {x ∈ Zn+1 : xi = 0}

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples: Integer lattice Zn. Checkerboard lattice Dn = {x ∈ Zn : xi even } Simplex lattice An = {x ∈ Zn+1 : xi = 0} Special root lattices E6, E7, E8.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples: Integer lattice Zn. Checkerboard lattice Dn = {x ∈ Zn : xi even } Simplex lattice An = {x ∈ Zn+1 : xi = 0} Special root lattices E6, E7, E8.

E8 generated by D8 and all-halves vector.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples: Integer lattice Zn. Checkerboard lattice Dn = {x ∈ Zn : xi even } Simplex lattice An = {x ∈ Zn+1 : xi = 0} Special root lattices E6, E7, E8.

E8 generated by D8 and all-halves vector. E7 orthogonal complement of a root (or A1) in E8.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Lattices

Definition

A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn. Examples: Integer lattice Zn. Checkerboard lattice Dn = {x ∈ Zn : xi even } Simplex lattice An = {x ∈ Zn+1 : xi = 0} Special root lattices E6, E7, E8.

E8 generated by D8 and all-halves vector. E7 orthogonal complement of a root (or A1) in E8. E6 orthogonal complement of an A2 in E8.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 7 / 47

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Projection of E8 root system

Image: Jgmoxness (Wikipedia), CC BY-SA 3.0 license Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 8 / 47

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Leech lattice

In dimension 24, there is also the remarkable Leech lattice. It is the unique even unimodular lattice in that dimension without any roots.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 9 / 47

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Leech lattice

In dimension 24, there is also the remarkable Leech lattice. It is the unique even unimodular lattice in that dimension without any roots. There are many neat constructions of it (for instance, Conway-Sloane give twenty-three constructions). The usual one involves the extended Golay code.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 9 / 47

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Leech lattice

In dimension 24, there is also the remarkable Leech lattice. It is the unique even unimodular lattice in that dimension without any roots. There are many neat constructions of it (for instance, Conway-Sloane give twenty-three constructions). The usual one involves the extended Golay code. My favorite: The lattice II25,1 is generated in R25,1 (which has the quadratic form x2

1 + · · · + x2 25 − x2 26) by vectors in Z26 or (Z + 1/2)26 with

even coordinate sum.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 9 / 47

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Leech lattice

In dimension 24, there is also the remarkable Leech lattice. It is the unique even unimodular lattice in that dimension without any roots. There are many neat constructions of it (for instance, Conway-Sloane give twenty-three constructions). The usual one involves the extended Golay code. My favorite: The lattice II25,1 is generated in R25,1 (which has the quadratic form x2

1 + · · · + x2 25 − x2 26) by vectors in Z26 or (Z + 1/2)26 with

even coordinate sum. The Weyl vector w = (0, 1, 2, . . . , 24, 70) has norm 0, since 12 + · · · + 242 = 702 (!)

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 9 / 47

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Leech lattice

In dimension 24, there is also the remarkable Leech lattice. It is the unique even unimodular lattice in that dimension without any roots. There are many neat constructions of it (for instance, Conway-Sloane give twenty-three constructions). The usual one involves the extended Golay code. My favorite: The lattice II25,1 is generated in R25,1 (which has the quadratic form x2

1 + · · · + x2 25 − x2 26) by vectors in Z26 or (Z + 1/2)26 with

even coordinate sum. The Weyl vector w = (0, 1, 2, . . . , 24, 70) has norm 0, since 12 + · · · + 242 = 702 (!) The Leech lattice is w⊥/Zw with the induced quadratic form.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 9 / 47

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Lattice packing

Associated sphere packing: if m(Λ) is the length of a smallest non-zero vector of Λ, then we can put balls of radius m(Λ)/2 around each point of Λ so that they don’t overlap.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 10 / 47

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Lattice packing

Associated sphere packing: if m(Λ) is the length of a smallest non-zero vector of Λ, then we can put balls of radius m(Λ)/2 around each point of Λ so that they don’t overlap. The packing problem for lattices asks for the densest lattice(s) in Rn for every n. This is equivalent to the determination of the Hermite constant γn, which arises in the geometry of numbers. The known answers are: n 1 2 3 4 5 6 7 8 24 Λ A1 A2 A3 D4 D5 E6 E7 E8 Leech due to Lagrange Gauss Korkine- Blichfeldt Cohn- Zolotareff Kumar

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 10 / 47

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The new results

Theorem (Viazovska)

The E8 lattice packing is the densest sphere packing in R8.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 11 / 47

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The new results

Theorem (Viazovska)

The E8 lattice packing is the densest sphere packing in R8.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 11 / 47

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The new results

Theorem (Viazovska)

The E8 lattice packing is the densest sphere packing in R8.

Theorem (Cohn-Kumar-Miller-Radchenko-Viazovska)

The Leech lattice packing is the densest sphere packing in R24.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 11 / 47

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The new results

Theorem (Viazovska)

The E8 lattice packing is the densest sphere packing in R8.

Theorem (Cohn-Kumar-Miller-Radchenko-Viazovska)

The Leech lattice packing is the densest sphere packing in R24.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 11 / 47

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The new results

Theorem (Viazovska)

The E8 lattice packing is the densest sphere packing in R8.

Theorem (Cohn-Kumar-Miller-Radchenko-Viazovska)

The Leech lattice packing is the densest sphere packing in R24. The proof is fairly direct, using just two main ingredients:

1 linear programming bounds for packing Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 11 / 47

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The new results

Theorem (Viazovska)

The E8 lattice packing is the densest sphere packing in R8.

Theorem (Cohn-Kumar-Miller-Radchenko-Viazovska)

The Leech lattice packing is the densest sphere packing in R24. The proof is fairly direct, using just two main ingredients:

1 linear programming bounds for packing 2 the theory of modular forms Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 11 / 47

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Linear programming bounds

Let the Fourier transform of a function f be defined by ˆ f (t) =

  • Rn f (x)e2πix,tdx.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 12 / 47

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Linear programming bounds

Let the Fourier transform of a function f be defined by ˆ f (t) =

  • Rn f (x)e2πix,tdx.

Theorem (Cohn-Elkies)

Suppose f : Rn → R is a Schwartz function with the properties

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 12 / 47

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Linear programming bounds

Let the Fourier transform of a function f be defined by ˆ f (t) =

  • Rn f (x)e2πix,tdx.

Theorem (Cohn-Elkies)

Suppose f : Rn → R is a Schwartz function with the properties

1 f (0) = ˆ

f (0) = 1.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 12 / 47

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Linear programming bounds

Let the Fourier transform of a function f be defined by ˆ f (t) =

  • Rn f (x)e2πix,tdx.

Theorem (Cohn-Elkies)

Suppose f : Rn → R is a Schwartz function with the properties

1 f (0) = ˆ

f (0) = 1.

2 f (x) ≤ 0 for |x| ≥ r (for some number r > 0). Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 12 / 47

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SLIDE 37

Linear programming bounds

Let the Fourier transform of a function f be defined by ˆ f (t) =

  • Rn f (x)e2πix,tdx.

Theorem (Cohn-Elkies)

Suppose f : Rn → R is a Schwartz function with the properties

1 f (0) = ˆ

f (0) = 1.

2 f (x) ≤ 0 for |x| ≥ r (for some number r > 0). 3

ˆ f (t) ≥ 0 for all t.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 12 / 47

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Linear programming bounds

Let the Fourier transform of a function f be defined by ˆ f (t) =

  • Rn f (x)e2πix,tdx.

Theorem (Cohn-Elkies)

Suppose f : Rn → R is a Schwartz function with the properties

1 f (0) = ˆ

f (0) = 1.

2 f (x) ≤ 0 for |x| ≥ r (for some number r > 0). 3

ˆ f (t) ≥ 0 for all t.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 12 / 47

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SLIDE 39

Linear programming bounds

Let the Fourier transform of a function f be defined by ˆ f (t) =

  • Rn f (x)e2πix,tdx.

Theorem (Cohn-Elkies)

Suppose f : Rn → R is a Schwartz function with the properties

1 f (0) = ˆ

f (0) = 1.

2 f (x) ≤ 0 for |x| ≥ r (for some number r > 0). 3

ˆ f (t) ≥ 0 for all t. Then the density of any sphere packing in Rn is bounded above by vol(Bn)(r/2)n.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 12 / 47

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LP bounds

Why is it a linear programming bound?

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 13 / 47

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LP bounds

Why is it a linear programming bound? We can rephrase the theorem (by scaling f appropriately) to say: if

1

ˆ f (0) = 1

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 13 / 47

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LP bounds

Why is it a linear programming bound? We can rephrase the theorem (by scaling f appropriately) to say: if

1

ˆ f (0) = 1

2 f (x) ≤ 0 for |x| ≥ 1 Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 13 / 47

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LP bounds

Why is it a linear programming bound? We can rephrase the theorem (by scaling f appropriately) to say: if

1

ˆ f (0) = 1

2 f (x) ≤ 0 for |x| ≥ 1 3

ˆ f (t) ≥ 0 for all t

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 13 / 47

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SLIDE 44

LP bounds

Why is it a linear programming bound? We can rephrase the theorem (by scaling f appropriately) to say: if

1

ˆ f (0) = 1

2 f (x) ≤ 0 for |x| ≥ 1 3

ˆ f (t) ≥ 0 for all t

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 13 / 47

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SLIDE 45

LP bounds

Why is it a linear programming bound? We can rephrase the theorem (by scaling f appropriately) to say: if

1

ˆ f (0) = 1

2 f (x) ≤ 0 for |x| ≥ 1 3

ˆ f (t) ≥ 0 for all t then the density is bounded by 2−nvol(Bn)f (0).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 13 / 47

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SLIDE 46

LP bounds

Why is it a linear programming bound? We can rephrase the theorem (by scaling f appropriately) to say: if

1

ˆ f (0) = 1

2 f (x) ≤ 0 for |x| ≥ 1 3

ˆ f (t) ≥ 0 for all t then the density is bounded by 2−nvol(Bn)f (0). Note that the constraints and objective function given are linear in f . Therefore this is a linear (convex) program.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 13 / 47

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Proof

Let’s see the proof for lattices: Let Λ be any lattice, which we have scaled so its minimal nonzero vector length is 1. Then the Poisson summation formula tells us

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 14 / 47

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Proof

Let’s see the proof for lattices: Let Λ be any lattice, which we have scaled so its minimal nonzero vector length is 1. Then the Poisson summation formula tells us

  • x∈Λ

f (x) = 1 covol(Λ)

  • t∈Λ∗

ˆ f (t)

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 14 / 47

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Proof

Let’s see the proof for lattices: Let Λ be any lattice, which we have scaled so its minimal nonzero vector length is 1. Then the Poisson summation formula tells us

  • x∈Λ

f (x) = 1 covol(Λ)

  • t∈Λ∗

ˆ f (t) Now the LHS is ≤ f (0) while the sum in the RHS is ≥ ˆ f (0) ≥ 1, yielding 1 covol(Λ) ≤ f (0) multiplying by the volume of a ball of radius 1/2 tells us that the density is at most 2−nvol(Bn)f (0).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 14 / 47

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Remarks on the LP bound

We can assume f is radial without loss of generality.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 15 / 47

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Remarks on the LP bound

We can assume f is radial without loss of generality. For numerical experimentation we can take f (x) =

N

  • i=0

ciLi(2π|x|2) exp(−π|x|2) where ci are the coefficients of the linear program, Li are the Laguerre polynomials (so Li times Gaussian is an eigenfunction for the Fourier transform).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 15 / 47

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Remarks on the LP bound

We can assume f is radial without loss of generality. For numerical experimentation we can take f (x) =

N

  • i=0

ciLi(2π|x|2) exp(−π|x|2) where ci are the coefficients of the linear program, Li are the Laguerre polynomials (so Li times Gaussian is an eigenfunction for the Fourier transform). In dimensions 8 and 24 one can get upper bounds which are numerically very close to the lower bound coming from E8 or Leech density.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 15 / 47

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LP bounds with dimension

Here is a plot of log(density) vs. dimension.

4 8 12 16 20 24 28 32

  • 14
  • 13
  • 12
  • 11
  • 10
  • 9
  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3

Rogers upper bound Cohn-Elkies upper bound Best packing known Look at slopes (asymptotically) as well as where these curves meet.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 16 / 47

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Desired functions

Let Λ be E8 or the Leech lattice, and r0, r1, . . . its nonzero vector lengths (square roots of the even natural numbers, except Leech skips 2). To have a tight upper bound that matches Λ, we need the function f to look like this:

r0 r1 r2

x f(x)

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 17 / 47

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SLIDE 55

Desired functions

While ˆ f must look like this:

r0 r1 r2

x f(x)

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 18 / 47

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SLIDE 56

Impasse

In [Cohn-Kumar 2009] we used a polynomial of degree 803 and 3000 digits

  • f precision to find f and ˆ

f which looked like this with 200 forced double roots, and r very close to 2.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 19 / 47

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SLIDE 57

Impasse

In [Cohn-Kumar 2009] we used a polynomial of degree 803 and 3000 digits

  • f precision to find f and ˆ

f which looked like this with 200 forced double roots, and r very close to 2. Obtained an upper bound of Leech lattice density times 1 + 10−30. Similar bounds for E8. Enough to show Λ24 is the densest lattice.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 19 / 47

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SLIDE 58

Impasse

In [Cohn-Kumar 2009] we used a polynomial of degree 803 and 3000 digits

  • f precision to find f and ˆ

f which looked like this with 200 forced double roots, and r very close to 2. Obtained an upper bound of Leech lattice density times 1 + 10−30. Similar bounds for E8. Enough to show Λ24 is the densest lattice. Further numerical experimentation by Cohn and Miller.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 19 / 47

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SLIDE 59

Impasse

In [Cohn-Kumar 2009] we used a polynomial of degree 803 and 3000 digits

  • f precision to find f and ˆ

f which looked like this with 200 forced double roots, and r very close to 2. Obtained an upper bound of Leech lattice density times 1 + 10−30. Similar bounds for E8. Enough to show Λ24 is the densest lattice. Further numerical experimentation by Cohn and Miller. But how do we write down exact functions??

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 19 / 47

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SLIDE 60

Impasse

In [Cohn-Kumar 2009] we used a polynomial of degree 803 and 3000 digits

  • f precision to find f and ˆ

f which looked like this with 200 forced double roots, and r very close to 2. Obtained an upper bound of Leech lattice density times 1 + 10−30. Similar bounds for E8. Enough to show Λ24 is the densest lattice. Further numerical experimentation by Cohn and Miller. But how do we write down exact functions?? We were stuck for more than a decade.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 19 / 47

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SLIDE 61

Enter Viazovska + modular forms

In March 2016 Viazovska posted a preprint to the arxiv, solving the sphere packing problem in 8 dimensions. She found the magic function f !

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 20 / 47

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SLIDE 62

Enter Viazovska + modular forms

In March 2016 Viazovska posted a preprint to the arxiv, solving the sphere packing problem in 8 dimensions. She found the magic function f ! Her proof used modular forms.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 20 / 47

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SLIDE 63

Modular group

A modular form is a function φ : H → C with a lot of symmetries.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 21 / 47

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SLIDE 64

Modular group

A modular form is a function φ : H → C with a lot of symmetries. Specifically, let SL2(Z) denote all the integer two by two matrices of determinant 1. It acts on the upper half plane by fractional linear transformations: a b c d

  • · z = az + b

cz + d

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 21 / 47

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SLIDE 65

Modular group

A modular form is a function φ : H → C with a lot of symmetries. Specifically, let SL2(Z) denote all the integer two by two matrices of determinant 1. It acts on the upper half plane by fractional linear transformations: a b c d

  • · z = az + b

cz + d In fact the action factors through PSL2(Z) = SL2(Z)/{±1}, and this quotient group is generated by the images of S = −1 1

  • and T =

1 1 1

  • .

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 21 / 47

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SLIDE 66

Fundamental domain

The picture shows Dedekind’s famous tesselation of the upper half plane. The union of a black and a white region makes a fundamental domain for the action of SL2(Z).

Image from the blog neverendingbooks.org, originally from John Stillwell’s article “Modular miracles” in Amer. Math. Monthly. Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 22 / 47

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SLIDE 67

Modular curves

The quotient SL2(Z)\H can be identified with the Riemann sphere CP1 minus a point. Compactifying the quotient by adding this cusp gives an algebraic curve (namely CP1). The preimages of this point are ∞ and the rational numbers, i.e. P1(Q).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 23 / 47

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SLIDE 68

Modular curves

The quotient SL2(Z)\H can be identified with the Riemann sphere CP1 minus a point. Compactifying the quotient by adding this cusp gives an algebraic curve (namely CP1). The preimages of this point are ∞ and the rational numbers, i.e. P1(Q). The principal congruence subgroup of level N is the subgroup Γ(N) of all the elements of SL2(Z) congruent to the identity modulo N. We say Γ is a congruence subgroup if it contains some Γ(N). Again the quotient is a complex algebraic curve; we can compactify it by adding finitely many cusps, which correpond to the elements of Γ\P1(Q).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 23 / 47

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SLIDE 69

Modular forms

The first condition for a holomorphic function f : H → C to be a modular form for Γ of weight k is f az + b cz + d

  • = (cz + d)kf (z)

for all matrices g = a b c d

  • ∈ Γ.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 24 / 47

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SLIDE 70

Modular forms

The first condition for a holomorphic function f : H → C to be a modular form for Γ of weight k is f az + b cz + d

  • = (cz + d)kf (z)

for all matrices g = a b c d

  • ∈ Γ.

Now, for some N the matrix 1 N 1

  • lies in the congruence subgroup, so we must have f (z + N) = f (z).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 24 / 47

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SLIDE 71

Growth condition

So if q = exp(2πiz) then we can write f as a function of q1/N. The second condition for a modular form says that near ∞, there is a power series expansion f =

  • n≥0

anqn/N.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 25 / 47

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SLIDE 72

Growth condition

So if q = exp(2πiz) then we can write f as a function of q1/N. The second condition for a modular form says that near ∞, there is a power series expansion f =

  • n≥0

anqn/N. Similarly for all the (finitely many) cusps. Defining the slash operator for g ∈ SL2(Z) as above by (f |kg)(z) = (cz + d)−kf (gz), all these f |kg must have holomorphic power series expansion at ∞.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 25 / 47

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SLIDE 73

Growth condition

So if q = exp(2πiz) then we can write f as a function of q1/N. The second condition for a modular form says that near ∞, there is a power series expansion f =

  • n≥0

anqn/N. Similarly for all the (finitely many) cusps. Defining the slash operator for g ∈ SL2(Z) as above by (f |kg)(z) = (cz + d)−kf (gz), all these f |kg must have holomorphic power series expansion at ∞. If it’s only a Laurent series, i.e., there are (finitely many) negative powers

  • f q, we say that f is a weakly holomorphic modular form.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 25 / 47

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SLIDE 74

Examples

How do we find actual examples of modular forms?

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 26 / 47

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SLIDE 75

Examples

How do we find actual examples of modular forms? The first way is to take simple examples of a “well-behaved” holomorphic function and symmetrize (recalling that SL2(Z) acts on Z2): Gk(z) =

  • (a,b)∈Z2\(0,0)

1 (az + b)k .

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 26 / 47

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SLIDE 76

Examples

How do we find actual examples of modular forms? The first way is to take simple examples of a “well-behaved” holomorphic function and symmetrize (recalling that SL2(Z) acts on Z2): Gk(z) =

  • (a,b)∈Z2\(0,0)

1 (az + b)k . For even k ≥ 4, the sum converges absolutely and we get a non-zero modular form of weight k. These are called Eisenstein series.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 26 / 47

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SLIDE 77

Eisenstein series

The normalized versions are E4 = 1 + 240

  • σ3(n)qn

E6 = 1 − 504

  • σ5(n)qn

Here σk(n) =

d|n,d>0 dk.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 27 / 47

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SLIDE 78

Eisenstein series

The normalized versions are E4 = 1 + 240

  • σ3(n)qn

E6 = 1 − 504

  • σ5(n)qn

Here σk(n) =

d|n,d>0 dk.

These two in fact generate the algebra of modular forms for the full modular group SL2(Z).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 27 / 47

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SLIDE 79

Eisenstein series

The normalized versions are E4 = 1 + 240

  • σ3(n)qn

E6 = 1 − 504

  • σ5(n)qn

Here σk(n) =

d|n,d>0 dk.

These two in fact generate the algebra of modular forms for the full modular group SL2(Z). Another beautiful example is the modular discriminant of weight 12 ∆ = (E 3

4 − E 2 6 )/1728 = q ∞

  • n=1

(1 − qn)24

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 27 / 47

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SLIDE 80

Theta functions

Another source of modular forms is theta functions of lattices: If Λ is an integral lattice (i.e. all inner products between vectors in the lattice are integers) of dimension d then ΘΛ(q) =

  • v∈Λ

qv,v/2 =

  • n≥0

Nn(Λ)qn/2 is a modular form of weight d/2 for some congruence subgroup (related to covol(Λ)).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 28 / 47

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SLIDE 81

Theta functions

Another source of modular forms is theta functions of lattices: If Λ is an integral lattice (i.e. all inner products between vectors in the lattice are integers) of dimension d then ΘΛ(q) =

  • v∈Λ

qv,v/2 =

  • n≥0

Nn(Λ)qn/2 is a modular form of weight d/2 for some congruence subgroup (related to covol(Λ)).

Example

The theta function of E8 is the Eisenstein series E4!

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 28 / 47

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SLIDE 82

Theta functions II

There are also classical theta functions studied by Jacobi, of which we will need: Θ00(z) :=

  • n∈Z

exp(πin2z) (the theta function of Z)

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 29 / 47

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SLIDE 83

Theta functions II

There are also classical theta functions studied by Jacobi, of which we will need: Θ00(z) :=

  • n∈Z

exp(πin2z) (the theta function of Z) Θ01(z) :=

  • n∈Z

(−1)n exp(πin2z) Θ10(z) :=

  • n∈Z

exp(πi(n + 1/2)2z) Let U = Θ4

00, V = Θ4 10, W = Θ4

  • 01. These are modular forms of weight 2

for the congruence subgroup Γ(2).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 29 / 47

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SLIDE 84

Theta functions II

There are also classical theta functions studied by Jacobi, of which we will need: Θ00(z) :=

  • n∈Z

exp(πin2z) (the theta function of Z) Θ01(z) :=

  • n∈Z

(−1)n exp(πin2z) Θ10(z) :=

  • n∈Z

exp(πi(n + 1/2)2z) Let U = Θ4

00, V = Θ4 10, W = Θ4

  • 01. These are modular forms of weight 2

for the congruence subgroup Γ(2). They are related by U = V + W .

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 29 / 47

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SLIDE 85

L-functions

Usually, from a modular form we make an L-function by taking a Mellin transform: L(f , s) = (2π)s Γ(s) ∞ f (it)ts dt t which works for ℜ(s) large enough.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 30 / 47

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SLIDE 86

L-functions

Usually, from a modular form we make an L-function by taking a Mellin transform: L(f , s) = (2π)s Γ(s) ∞ f (it)ts dt t which works for ℜ(s) large enough. These L-functions are a cornerstone of much of modern number theory.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 30 / 47

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SLIDE 87

L-functions

Usually, from a modular form we make an L-function by taking a Mellin transform: L(f , s) = (2π)s Γ(s) ∞ f (it)ts dt t which works for ℜ(s) large enough. These L-functions are a cornerstone of much of modern number theory. For instance, Wiles’s proof of FLT relies on showing the L-function of a specific kind of elliptic curve is the same as that of a modular form.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 30 / 47

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SLIDE 88

Quasimodular forms

If we apply the Eisenstein series construction to k = 2, we run into problems because of non-absolute convergence.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 31 / 47

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SLIDE 89

Quasimodular forms

If we apply the Eisenstein series construction to k = 2, we run into problems because of non-absolute convergence. However, we can define G2(z) =

  • n=0

1 n2 +

  • m=0
  • n∈Z

1 (mz + n)2 and this double sum converges. Normalizing we have E2 = 1 − 24

  • n≥0

σ1(n)qn.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 31 / 47

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SLIDE 90

Quasimodular forms

If we apply the Eisenstein series construction to k = 2, we run into problems because of non-absolute convergence. However, we can define G2(z) =

  • n=0

1 n2 +

  • m=0
  • n∈Z

1 (mz + n)2 and this double sum converges. Normalizing we have E2 = 1 − 24

  • n≥0

σ1(n)qn. The only problem is that E2 is not a genuine modular form: E2(−1/z) = z2E2(z) − 6i π z.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 31 / 47

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SLIDE 91

Quasimodular forms II

Together with modular forms, E2 generates the algebra of quasi-modular forms.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 32 / 47

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SLIDE 92

Quasimodular forms II

Together with modular forms, E2 generates the algebra of quasi-modular forms. It can also be obtained by differentiating modular forms. For f (z) =

  • anqn with q = exp(2πiz),

define f ′(z) := (Df )(z) := q df dq = 1 2πi df (z) dz .

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 32 / 47

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SLIDE 93

Quasimodular forms II

Together with modular forms, E2 generates the algebra of quasi-modular forms. It can also be obtained by differentiating modular forms. For f (z) =

  • anqn with q = exp(2πiz),

define f ′(z) := (Df )(z) := q df dq = 1 2πi df (z) dz . Then one can check E ′

4 = (E2E4 − E6)/3 and E ′ 6 = (E2E6 − E 2 4 )/2.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 32 / 47

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SLIDE 94

Quasimodular forms II

Together with modular forms, E2 generates the algebra of quasi-modular forms. It can also be obtained by differentiating modular forms. For f (z) =

  • anqn with q = exp(2πiz),

define f ′(z) := (Df )(z) := q df dq = 1 2πi df (z) dz . Then one can check E ′

4 = (E2E4 − E6)/3 and E ′ 6 = (E2E6 − E 2 4 )/2.

In general differentiating a weight k modular forms of weight ℓ times yields a polynomial in E2 of degree ℓ, and the resulting quasimodular form has weight k + 2ℓ. We call ℓ the depth of the quasimodular form.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 32 / 47

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SLIDE 95

Even eigenfunction

The magic functions for sphere packing arise as (Laplace) transforms of weakly holomorphic modular or quasi-modular forms.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 33 / 47

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SLIDE 96

Even eigenfunction

The magic functions for sphere packing arise as (Laplace) transforms of weakly holomorphic modular or quasi-modular forms. Consider the weakly holomorphic quasi-modular form of depth 2 φ0 = (E4E2 − E6)2 ∆

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 33 / 47

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SLIDE 97

Even eigenfunction

The magic functions for sphere packing arise as (Laplace) transforms of weakly holomorphic modular or quasi-modular forms. Consider the weakly holomorphic quasi-modular form of depth 2 φ0 = (E4E2 − E6)2 ∆ and for r > √ 2, define a(r) = −4 sin(πr2/2)2 i∞ φ0 −1 z

  • z2eπir2zdz.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 33 / 47

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SLIDE 98

Even eigenfunction

The magic functions for sphere packing arise as (Laplace) transforms of weakly holomorphic modular or quasi-modular forms. Consider the weakly holomorphic quasi-modular form of depth 2 φ0 = (E4E2 − E6)2 ∆ and for r > √ 2, define a(r) = −4 sin(πr2/2)2 i∞ φ0 −1 z

  • z2eπir2zdz.

We can extend give an alternative expression for the integral which extends the domain of definition to r > 0.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 33 / 47

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SLIDE 99

Even eigenfunction II

Note that: φ0(−1/(it))t2 = O(exp(2πt)) as t → ∞. So the integral has a term proportional to ∞ exp(−π(r2 − 2)t)dt = 1 π(r2 − 2) which downgrades the double zero of sin(πr2/2)2 to a single zero, as we wanted.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 34 / 47

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SLIDE 100

Even eigenfunction II

Note that: φ0(−1/(it))t2 = O(exp(2πt)) as t → ∞. So the integral has a term proportional to ∞ exp(−π(r2 − 2)t)dt = 1 π(r2 − 2) which downgrades the double zero of sin(πr2/2)2 to a single zero, as we wanted. The quasi-modular property of φ0 can be used to show that a(r) is an even eigenfunction: the Fourier transform replaces eπir2z by z−4eπir2(−1/z) and then we can use transformation properties under z → −1/z.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 34 / 47

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SLIDE 101

Even eigenfunction III

Write −4 sin2(πr2/2) = −2(1 − cos(πr2)) = exp(πir2) + exp(−πir2) − 2.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 35 / 47

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SLIDE 102

Even eigenfunction III

Write −4 sin2(πr2/2) = −2(1 − cos(πr2)) = exp(πir2) + exp(−πir2) − 2.

So a(r) = i∞ φ0(−1/z)z2 eπir 2(z+1) + eπir 2(z−1) − 2eπir 2z dz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 35 / 47

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SLIDE 103

Even eigenfunction III

Write −4 sin2(πr2/2) = −2(1 − cos(πr2)) = exp(πir2) + exp(−πir2) − 2.

So a(r) = i∞ φ0(−1/z)z2 eπir 2(z+1) + eπir 2(z−1) − 2eπir 2z dz = i∞ φ0(−1/z)z2eπir 2(z+1)dz + i∞ φ0(−1/z)z2eπir 2(z−1)dz − 2 i∞ φ0(−1/z)z2eπir 2dz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 35 / 47

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SLIDE 104

Even eigenfunction III

Write −4 sin2(πr2/2) = −2(1 − cos(πr2)) = exp(πir2) + exp(−πir2) − 2.

So a(r) = i∞ φ0(−1/z)z2 eπir 2(z+1) + eπir 2(z−1) − 2eπir 2z dz = i∞ φ0(−1/z)z2eπir 2(z+1)dz + i∞ φ0(−1/z)z2eπir 2(z−1)dz − 2 i∞ φ0(−1/z)z2eπir 2dz = i∞+1

1

φ0 −1 u − 1

  • (u − 1)2eπir 2udu +

i∞−1

−1

φ0 −1 u + 1

  • (u + 1)2eπir 2udu

− 2 i∞ φ0(−1/z)z2eπir 2zdu

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 35 / 47

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SLIDE 105

Even eigenfunction IV

We can shift the contour at infinity, and break up the path.

a(r) = i

1

φ0 −1 z − 1

  • (z − 1)2eπir 2zdz +

i∞

i

φ0 −1 z − 1

  • (z − 1)2eπir 2zdz

+ i

−1

φ0 −1 z + 1

  • (z + 1)2eπir 2zdz +

i∞

i

φ0 −1 z + 1

  • (z + 1)2eπir 2zdz

− 2 i φ0(−1/z)z2eπir 2dz − 2 i∞

i

φ0(−1/z)z2eπir 2dz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 36 / 47

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SLIDE 106

Even eigenfunction IV

We can shift the contour at infinity, and break up the path.

a(r) = i

1

φ0 −1 z − 1

  • (z − 1)2eπir 2zdz +

i∞

i

φ0 −1 z − 1

  • (z − 1)2eπir 2zdz

+ i

−1

φ0 −1 z + 1

  • (z + 1)2eπir 2zdz +

i∞

i

φ0 −1 z + 1

  • (z + 1)2eπir 2zdz

− 2 i φ0(−1/z)z2eπir 2dz − 2 i∞

i

φ0(−1/z)z2eπir 2dz

We will combine the second, fourth and sixth integrals. Note that z2φ0(−1/z) = z2φ0(z) + zφ−2(z) + φ−4(z) where φ0, φ−2, φ−4 are quasimodular forms of depth 2, 1, 0 and weight 0, −2, −4 respectively. In any case, they are all invariant under T.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 36 / 47

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SLIDE 107

Even eigenfunction V

Therefore, the second difference operator just acts on the multipliers on z2, z, 1, yielding

φ0 −1 z + 1

  • (z + 1)2 + φ0

−1 z − 1

  • (z − 1)2 − φ0

−1 z

  • z2

= φ0(z)

  • (z + 1)2 + (z − 1)2 − 2z2

= 2φ0(z).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 37 / 47

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SLIDE 108

Even eigenfunction V

Therefore, the second difference operator just acts on the multipliers on z2, z, 1, yielding

φ0 −1 z + 1

  • (z + 1)2 + φ0

−1 z − 1

  • (z − 1)2 − φ0

−1 z

  • z2

= φ0(z)

  • (z + 1)2 + (z − 1)2 − 2z2

= 2φ0(z).

Therefore

a(r) = i

1

φ0 −1 z − 1

  • (z − 1)2eπir 2zdz +

i

−1

φ0 −1 z + 1

  • (z + 1)2eπir 2zdz

− 2 i φ0(−1/z)z2eπir 2zdz + 2 i∞

i

2φ0(z)eπir 2zdz.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 37 / 47

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SLIDE 109

Fourier transform

We have

  • a(r) =

i

1

φ0

  • −1

z−1

(z − 1)2 z4 e

πir 2 −1 z

  • dz +

i

−1

φ0

  • −1

z+1

(z + 1)2 z4 e

πir 2 −1 z

  • dz

− 2 i φ0(−1/z)z2z−4eπir 2(−1/z)dz − 2 i∞

i

2φ0(z)z−4eπir 2(−1/z)dz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 38 / 47

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SLIDE 110

Fourier transform

We have

  • a(r) =

i

1

φ0

  • −1

z−1

(z − 1)2 z4 e

πir 2 −1 z

  • dz +

i

−1

φ0

  • −1

z+1

(z + 1)2 z4 e

πir 2 −1 z

  • dz

− 2 i φ0(−1/z)z2z−4eπir 2(−1/z)dz − 2 i∞

i

2φ0(z)z−4eπir 2(−1/z)dz = i

−1

φ0

  • 1 −

1 w+1

  • (w + 1)2eπir 2wdw +

i

1

φ0

  • −1

w−1 − 1

  • (w − 1)2eπir 2wdw

− 2 i

i∞

φ0(w)eπir 2wdw + 2

i

2φ0(−1/w)w 2eπir 2wdw

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 38 / 47

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SLIDE 111

Fourier transform

We have

  • a(r) =

i

1

φ0

  • −1

z−1

(z − 1)2 z4 e

πir 2 −1 z

  • dz +

i

−1

φ0

  • −1

z+1

(z + 1)2 z4 e

πir 2 −1 z

  • dz

− 2 i φ0(−1/z)z2z−4eπir 2(−1/z)dz − 2 i∞

i

2φ0(z)z−4eπir 2(−1/z)dz = i

−1

φ0

  • 1 −

1 w+1

  • (w + 1)2eπir 2wdw +

i

1

φ0

  • −1

w−1 − 1

  • (w − 1)2eπir 2wdw

− 2 i

i∞

φ0(w)eπir 2wdw + 2

i

2φ0(−1/w)w 2eπir 2wdw = a(r). using the change of variable z = −1/w, dz = 1/w 2dw, and the T-invariance of φ0.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 38 / 47

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SLIDE 112

Fourier transform

We have

  • a(r) =

i

1

φ0

  • −1

z−1

(z − 1)2 z4 e

πir 2 −1 z

  • dz +

i

−1

φ0

  • −1

z+1

(z + 1)2 z4 e

πir 2 −1 z

  • dz

− 2 i φ0(−1/z)z2z−4eπir 2(−1/z)dz − 2 i∞

i

2φ0(z)z−4eπir 2(−1/z)dz = i

−1

φ0

  • 1 −

1 w+1

  • (w + 1)2eπir 2wdw +

i

1

φ0

  • −1

w−1 − 1

  • (w − 1)2eπir 2wdw

− 2 i

i∞

φ0(w)eπir 2wdw + 2

i

2φ0(−1/w)w 2eπir 2wdw = a(r). using the change of variable z = −1/w, dz = 1/w 2dw, and the T-invariance of φ0. So we have created a +1-eigenfunction for the Fourier transform.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 38 / 47

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SLIDE 113

Odd eigenfunction

Let ψ = 2W 3(5UV + 2W 2) ∆ . It is a weakly holomorphic modular form of weight −2 for the congruence subgroup Γ0(2).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 39 / 47

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SLIDE 114

Odd eigenfunction

Let ψ = 2W 3(5UV + 2W 2) ∆ . It is a weakly holomorphic modular form of weight −2 for the congruence subgroup Γ0(2). Define b(r) = −4r2 sin(πr2/2)2 i∞ ψ(z)eπir2zdz.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 39 / 47

slide-115
SLIDE 115

Odd eigenfunction

Let ψ = 2W 3(5UV + 2W 2) ∆ . It is a weakly holomorphic modular form of weight −2 for the congruence subgroup Γ0(2). Define b(r) = −4r2 sin(πr2/2)2 i∞ ψ(z)eπir2zdz. We can similarly show that b(r) is an odd eigenfunction for the Fourier transform, and has a single root at r = √ 2 and double roots at other √ 2n.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 39 / 47

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SLIDE 116

Odd eigenfunction II

Write ψT = ψ|−2T and ψS = ψ|−2S. Then it is easy to verify that ψS + ψT = ψ, from which it follows that ψT|−2S = −ψT. Also, ψS|−2S = ψ and finally ψ|−2T −1 = ψT since T −2 ∈ Γ(2).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 40 / 47

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SLIDE 117

Odd eigenfunction II

Write ψT = ψ|−2T and ψS = ψ|−2S. Then it is easy to verify that ψS + ψT = ψ, from which it follows that ψT|−2S = −ψT. Also, ψS|−2S = ψ and finally ψ|−2T −1 = ψT since T −2 ∈ Γ(2). We rewrite the integral as before b(r) = i∞ ψ(z)eπir2(z+1)dz + i∞ ψ(z)eπir2(z−1)dz − 2 i∞ ψ(z)eπir2zdz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 40 / 47

slide-118
SLIDE 118

Odd eigenfunction II

Write ψT = ψ|−2T and ψS = ψ|−2S. Then it is easy to verify that ψS + ψT = ψ, from which it follows that ψT|−2S = −ψT. Also, ψS|−2S = ψ and finally ψ|−2T −1 = ψT since T −2 ∈ Γ(2). We rewrite the integral as before b(r) = i∞ ψ(z)eπir2(z+1)dz + i∞ ψ(z)eπir2(z−1)dz − 2 i∞ ψ(z)eπir2zdz = i∞

1

ψ(z − 1)eπir2zdz + i∞

−1

ψ(z + 1)eπir2zdz − 2 i∞ ψ(z)eπir2zdz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 40 / 47

slide-119
SLIDE 119

Odd eigenfunction II

Write ψT = ψ|−2T and ψS = ψ|−2S. Then it is easy to verify that ψS + ψT = ψ, from which it follows that ψT|−2S = −ψT. Also, ψS|−2S = ψ and finally ψ|−2T −1 = ψT since T −2 ∈ Γ(2). We rewrite the integral as before b(r) = i∞ ψ(z)eπir2(z+1)dz + i∞ ψ(z)eπir2(z−1)dz − 2 i∞ ψ(z)eπir2zdz = i∞

1

ψ(z − 1)eπir2zdz + i∞

−1

ψ(z + 1)eπir2zdz − 2 i∞ ψ(z)eπir2zdz = i∞

1

ψT(z)eπir2zdz + i∞

−1

ψT(z)eπir2zdz − 2 i∞ ψ(z)eπir2zdz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 40 / 47

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SLIDE 120

Odd eigenfunction III

b(r) = i

1

ψT(z)eπir2zdz + i

−1

ψT(z)eπir2zdz − 2 i ψ(z)eπir2zdz + 2 i∞

i

(ψT(z) − ψ(z))eπir2zdz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 41 / 47

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SLIDE 121

Odd eigenfunction III

b(r) = i

1

ψT(z)eπir2zdz + i

−1

ψT(z)eπir2zdz − 2 i ψ(z)eπir2zdz + 2 i∞

i

(ψT(z) − ψ(z))eπir2zdz = i

1

ψT(z)eπir2zdz + i

−1

ψT(z)eπir2zdz − 2 i ψ(z)eπir2zdz − 2 i∞

i

ψS(z)eπir2zdz.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 41 / 47

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SLIDE 122

Odd eigenfunction III

b(r) = i

1

ψT(z)eπir2zdz + i

−1

ψT(z)eπir2zdz − 2 i ψ(z)eπir2zdz + 2 i∞

i

(ψT(z) − ψ(z))eπir2zdz = i

1

ψT(z)eπir2zdz + i

−1

ψT(z)eπir2zdz − 2 i ψ(z)eπir2zdz − 2 i∞

i

ψS(z)eπir2zdz. This extends the domain of definition to r > 0. Note that ψ(it) = O(e2πt) as t → ∞ gives a pole at r = √ 2 for the integral, just as in the even case. To check that we have an odd eigenfunction, we compute

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 41 / 47

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SLIDE 123

Odd eigenfunction IV

  • b(r) =

i

1

ψT(z)z−4eπir2(−1/z)dz + i

−1

ψT(z)z−4eπir2(−1/z)dz − 2 i ψ(z)z−4eπir2(−1/z)dz − 2 i∞

i

ψS(z)z−4eπir2(−1/z)dz

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 42 / 47

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SLIDE 124

Odd eigenfunction IV

  • b(r) =

i

1

ψT(z)z−4eπir2(−1/z)dz + i

−1

ψT(z)z−4eπir2(−1/z)dz − 2 i ψ(z)z−4eπir2(−1/z)dz − 2 i∞

i

ψS(z)z−4eπir2(−1/z)dz = i

1

ψT(−1/w)w2eπir2wdw + i

−1

ψT(−1/w)w2eπir2wdw − 2 i ψ(−1/w)w2eπir2wdw − 2 i∞

i

ψS(−1/w)w2eπir2wdw

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 42 / 47

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SLIDE 125

Odd eigenfunction IV

  • b(r) =

i

1

ψT(z)z−4eπir2(−1/z)dz + i

−1

ψT(z)z−4eπir2(−1/z)dz − 2 i ψ(z)z−4eπir2(−1/z)dz − 2 i∞

i

ψS(z)z−4eπir2(−1/z)dz = i

1

ψT(−1/w)w2eπir2wdw + i

−1

ψT(−1/w)w2eπir2wdw − 2 i ψ(−1/w)w2eπir2wdw − 2 i∞

i

ψS(−1/w)w2eπir2wdw = i

−1

ψTS(w)eπir2wdw + i

1

ψTS(w)eπir2wdw − 2 i

∞

ψS(w)eπir2wdw − 2

i

ψ(w)eπir2wdw

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 42 / 47

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SLIDE 126

Odd eigenfunction V

So

  • b(r) = −

i

−1

ψT(w)eπir2wdw − i

1

ψT(w)eπir2wdw + 2 ∞

i

ψS(w)eπir2wdw + 2 i ψ(w)eπir2wdw = −b(r) where we used ψTS = −ψT.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 43 / 47

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SLIDE 127

Putting everything together

Now, we can take a linear combination of a(r) and b(r) to make f such that f and ˆ f have the desired properties (for instance, to make ˆ f vanish to

  • rder 2 at

√ 2.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 44 / 47

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SLIDE 128

Putting everything together

Now, we can take a linear combination of a(r) and b(r) to make f such that f and ˆ f have the desired properties (for instance, to make ˆ f vanish to

  • rder 2 at

√ 2. One still has to verify that there are no extra roots, but this can be done by analyzing the underlying integrands.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 44 / 47

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SLIDE 129

Putting everything together

Now, we can take a linear combination of a(r) and b(r) to make f such that f and ˆ f have the desired properties (for instance, to make ˆ f vanish to

  • rder 2 at

√ 2. One still has to verify that there are no extra roots, but this can be done by analyzing the underlying integrands. At the moment, this last verification of the required inequalities needs a computer-assisted proof.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 44 / 47

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SLIDE 130

Leech lattice

The proof of optimality of Leech in R24 proceeds along similar lines, though it is more complicated. We just write down the kernels here, which have the same form.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 45 / 47

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SLIDE 131

Leech lattice

The proof of optimality of Leech in R24 proceeds along similar lines, though it is more complicated. We just write down the kernels here, which have the same form. For the even eigenfunction, the integrand has the weakly holomorphic quasimodular form φ = (25E 4

4 − 49E 2 6 E4) + 48E6E 2 4 E2 + (−49E 3 4 + 25E 2 6 )E 2 2

∆2 .

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 45 / 47

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SLIDE 132

Leech lattice

The proof of optimality of Leech in R24 proceeds along similar lines, though it is more complicated. We just write down the kernels here, which have the same form. For the even eigenfunction, the integrand has the weakly holomorphic quasimodular form φ = (25E 4

4 − 49E 2 6 E4) + 48E6E 2 4 E2 + (−49E 3 4 + 25E 2 6 )E 2 2

∆2 . For the odd eigenfunction, the integrand has the weakly holomorphic modular form for Γ(2) ψ = W 5(7UV + 2W 2) ∆2 .

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 45 / 47

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SLIDE 133

Beyond sphere packing in 8 and 24 dimensions

One big open problem is to find magic functions for dimension 2 (even though we know the A2 lattice gives the densest sphere packing, by a relatively elementary argument).

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 46 / 47

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SLIDE 134

Beyond sphere packing in 8 and 24 dimensions

One big open problem is to find magic functions for dimension 2 (even though we know the A2 lattice gives the densest sphere packing, by a relatively elementary argument). In other dimensions, we do not expect this technique to give sharp bounds, but it may yield better upper bounds for sphere packing than the current records.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 46 / 47

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SLIDE 135

Beyond sphere packing in 8 and 24 dimensions

One big open problem is to find magic functions for dimension 2 (even though we know the A2 lattice gives the densest sphere packing, by a relatively elementary argument). In other dimensions, we do not expect this technique to give sharp bounds, but it may yield better upper bounds for sphere packing than the current records. We have since also worked on a wide generalization of the sphere packing problem to energy minimization, and have proved that E8 and the Leech lattice are universally optimal for Gaussian (and therefore inverse power law) potential functions in their respective dimensions, via sharp LP bounds for energy.

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 46 / 47

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SLIDE 136

References: Maryna Viazovska, “The sphere packing problem in dimension 8”, Ann. of Math 185 (2017), 991-1015. http://arxiv.org/abs/1603.04246 Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska, “The sphere packing problem in dimension 24”, Ann.

  • f Math 185 (2017), 1017-1033. http://arxiv.org/abs/1603.06518

David de Laat, Frank Vallentin, “A Breakthrough in Sphere Packing: The Search for Magic Functions”, Nieuw Archief voor Wiskunde (5) 17 (2016), 184-192. http://arxiv.org/abs/1607.02111 Henry Cohn, “A conceptual breakthrough in sphere packing”, Notices

  • Amer. Math. Soc. 64 (2017), no. 2, 102-115.

https://arxiv.org/abs/1611.01685

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 47 / 47

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SLIDE 137

References: Maryna Viazovska, “The sphere packing problem in dimension 8”, Ann. of Math 185 (2017), 991-1015. http://arxiv.org/abs/1603.04246 Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska, “The sphere packing problem in dimension 24”, Ann.

  • f Math 185 (2017), 1017-1033. http://arxiv.org/abs/1603.06518

David de Laat, Frank Vallentin, “A Breakthrough in Sphere Packing: The Search for Magic Functions”, Nieuw Archief voor Wiskunde (5) 17 (2016), 184-192. http://arxiv.org/abs/1607.02111 Henry Cohn, “A conceptual breakthrough in sphere packing”, Notices

  • Amer. Math. Soc. 64 (2017), no. 2, 102-115.

https://arxiv.org/abs/1611.01685

Thank you!

Abhinav Kumar (Stony Brook, ICTS) Recent breakthroughs in sphere packing November 8, 2019 47 / 47