[PPT] - 3-Coloring the Discrete Torus or Rigidity of zero temperature PowerPoint Presentation

SLIDE 1

Introduction Proof overview Ideas and Method Open problems

3-Coloring the Discrete Torus

r

Rigidity of zero temperature 3-states anti-ferromagnetic Potts model

Ohad N. Feldheim Joint work with Ron Peled

Institute for Mathematics and its Application

IMA PostDoc Seminar, October, 2014

SLIDE 2

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

3-Colorings of the Grid/Torus

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Zero boundary conditions

1 1 1 2 2 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Periodic boundary conditions

SLIDE 3

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Random 3-Colorings

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Zero boundary conditions

1 1 1 2 2 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Periodic boundary conditions

Uniformly chosen proper 3-coloring (Given boundary conditions) High dimension Zd, and Td

n.

SLIDE 4

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Additional Motivation

Physics Mathematical Physics Combinatorics

Antiferromagnetism q-states antiferromagnetic Potts model

2 1 2 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2

q-colorings of the discrete torus

1 1 1 2 2 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Generalizes the celebrated Ising model. Each point takes one of q values. Neighbors dislike getting the same color. 3-coloring is the “zero temperature” version.

SLIDE 5

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Properties of Interest

In a typical coloring: What is the typical relative frequency of the colors? Is it (1/3, 1/3, 1/3)?

SLIDE 6

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Properties of Interest

In a typical coloring: What is the typical relative frequency of the colors? Is it (1/3, 1/3, 1/3)? Does a typical coloring follow some pattern?

SLIDE 7

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Properties of Interest

In a typical coloring: What is the typical relative frequency of the colors? Is it (1/3, 1/3, 1/3)? Does a typical coloring follow some pattern? Is there long range correlation?

SLIDE 8

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Properties of Interest

In a typical coloring: What is the typical relative frequency of the colors? Is it (1/3, 1/3, 1/3)? Does a typical coloring follow some pattern? Is there long range correlation? Does it look roughly like this?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 9

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Properties of Interest

In a typical coloring: What is the typical relative frequency of the colors? Is it (1/3, 1/3, 1/3)? Does a typical coloring follow some pattern? Is there long range correlation? Does it look roughly like this?

Conjecture:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 10

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Properties of Interest

In a typical coloring: What is the typical relative frequency of the colors? Is it (1/3, 1/3, 1/3)? Does a typical coloring follow some pattern? Is there long range correlation? Does it look roughly like this?

Conjecture: d = 2 No.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 11

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Properties of Interest

In a typical coloring: What is the typical relative frequency of the colors? Is it (1/3, 1/3, 1/3)? Does a typical coloring follow some pattern? Is there long range correlation? Does it look roughly like this?

Conjecture: d = 2 No. d > 2 Yes.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 12

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Rigidity for 0-boundary

The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0.

SLIDE 13

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Rigidity for 0-boundary

The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0. Formally: Let d be large enough, a uniformly chosen 3-coloring with 0-BC, has: E |{v ∈ V even : g(v) = 0}| |V even| < exp

−

cd log2 d

.

SLIDE 14

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Rigidity for 0-boundary

The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0. Formally: Let d be large enough, a uniformly chosen 3-coloring with 0-BC, has: E |{v ∈ V even : g(v) = 0}| |V even| < exp

−

cd log2 d

.

Does not work for periodic BC.

SLIDE 15

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Rigidity for 0-boundary

The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0. Formally: Let d be large enough, a uniformly chosen 3-coloring with 0-BC, has: E |{v ∈ V even : g(v) = 0}| |V even| < exp

−

cd log2 d

.

Does not work for periodic BC. Open in low dimensions.

SLIDE 16

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Rigidity for the hypercube

The conjecture has also been supported on bounded tori. Periodic boundary on the even hypercube (Galvin & Engbers 2011) For every fixed n, for high enough dimension (depndeing on n), a typical 3-coloring with periodic boundary conditions is nearly constant on either the even or the odd sublattice.

SLIDE 17

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Rigidity for the hypercube

The conjecture has also been supported on bounded tori. Periodic boundary on the even hypercube (Galvin & Engbers 2011) For every fixed n, for high enough dimension (depndeing on n), a typical 3-coloring with periodic boundary conditions is nearly constant on either the even or the odd sublattice. Works also for q-colorings (and even more general!) Fixed n is less important for physicists.

SLIDE 18

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Some rigidity for the torus

Limited rigidity for periodic boundary (Galvin & Randall 2012, Galvin, Kahn, Randall, Sorkin 2014) For high enough dimension (depndeing on n), a typical 3-coloring with periodic boundary conditions has at least 0.22nd more zeroes

n one sublattice than on the other.

SLIDE 19

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Previous Results - Some rigidity for the torus

Limited rigidity for periodic boundary (Galvin & Randall 2012, Galvin, Kahn, Randall, Sorkin 2014) For high enough dimension (depndeing on n), a typical 3-coloring with periodic boundary conditions has at least 0.22nd more zeroes

n one sublattice than on the other.

Is not enough to show that one sublattice tends to be nearly monochromatic. Does not allow analysis of sloped colorings. Independent work and methods. More robust, and may be useful for non-zero temperatures.

SLIDE 20

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Our Results - Rigidity on Tn

d

We establish a parallel phenomenon for periodic BC. Theorem (F., Peled) A typical 3-coloring with periodic boundary conditions is nearly constant on either the even or the odd sublattice.

SLIDE 21

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Our Results - Rigidity on Tn

d

We establish a parallel phenomenon for periodic BC. Theorem (F., Peled) A typical 3-coloring with periodic boundary conditions is nearly constant on either the even or the odd sublattice. Formally: Let d be large enough, a uniformly chosen 3-coloring with periodic BC has: E    min

z∈{0,1,2}, s∈{even,odd}

|{v ∈ V s : g(v) = z}| |V s|    < exp

−

cd log2 d

.

SLIDE 22

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Our Results - Rigidity on Tn

d

We establish a parallel phenomenon for periodic BC. Theorem (F., Peled) A typical 3-coloring with periodic boundary conditions is nearly constant on either the even or the odd sublattice. Formally: Let d be large enough, a uniformly chosen 3-coloring with periodic BC has: E    min

z∈{0,1,2}, s∈{even,odd}

|{v ∈ V s : g(v) = z}| |V s|    < exp

−

cd log2 d

.

n must be even.

SLIDE 23

Introduction Proof overview Ideas and Method Open problems Coloring the Grid Previous results Our results

Our Results - Rigidity on Tn

d

We establish a parallel phenomenon for periodic BC. Theorem (F., Peled) A typical 3-coloring with periodic boundary conditions is nearly constant on either the even or the odd sublattice. Formally: Let d be large enough, a uniformly chosen 3-coloring with periodic BC has: E    min

z∈{0,1,2}, s∈{even,odd}

|{v ∈ V s : g(v) = z}| |V s|    < exp

−

cd log2 d

.

n must be even. Introduces topological techniques to the problem.

SLIDE 24

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Proof Overview

SLIDE 25

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Homomorphism Height Functions

h : G → Z satisfying |h(v) − h(u)| = 1 if v ∼ u.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2

1

2 2 2 2

1

2 2 2

1
1

2

1

3

Zero boundary conditions

3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2 2

1
1
1

2 2 2 2 2 2 2 2 2

Periodic boundary conditions

Discretized “topographical map”.

SLIDE 26

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Relation to 3-Colorings

On Zd there is a a natural bijection.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

Pointed 3-Colorings

mod 3 6 6 6 6 3 3 1 1 1 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

Pointed HHFs

SLIDE 27

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Relation to 3-Colorings

On Zd there is a a natural bijection.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

Pointed 3-Colorings

mod 3 6 6 6 6 3 3 1 1 1 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

Pointed HHFs

This bijection does not extend to Td

n.

SLIDE 28

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Rigidity of HHFs

More is known about HHFs then about 3-colorings: Rigidity of HHFs on Td

n (follows from Peled 2010)

A typical pointed HHF on a high dimensional torus is nearly constant on either the even or the odd sublattice.

SLIDE 29

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Rigidity of HHFs

More is known about HHFs then about 3-colorings: Rigidity of HHFs on Td

n (follows from Peled 2010)

A typical pointed HHF on a high dimensional torus is nearly constant on either the even or the odd sublattice. We wish to transfer this result to 3-colorings.

SLIDE 30

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Rigidity of HHFs

More is known about HHFs then about 3-colorings: Rigidity of HHFs on Td

n (follows from Peled 2010)

A typical pointed HHF on a high dimensional torus is nearly constant on either the even or the odd sublattice. We wish to transfer this result to 3-colorings. Main obstruction: No bijection between HHFs and colorings on Td

n.

SLIDE 31

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Rigidity of HHFs

More is known about HHFs then about 3-colorings: Rigidity of HHFs on Td

n (follows from Peled 2010)

A typical pointed HHF on a high dimensional torus is nearly constant on either the even or the odd sublattice. We wish to transfer this result to 3-colorings. Main obstruction: No bijection between HHFs and colorings on Td

n.

Here Topology enters.

SLIDE 32

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Introducing Quasi-Periodic HHFs What are 3-colorings on Td

n in bijection with?

SLIDE 33

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Introducing Quasi-Periodic HHFs

2

1

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

1

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

m1 = 6 m2 = 0

What are 3-colorings on Td

n in bijection with?

Quasi periodic HHFs of Zd

whose slopes are 0 mod 6 Td

n

Zd

SLIDE 34

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Flat Slope HHFs ↔ HHFs on Td

n

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 1 2 1 2 1 1 2 1 1 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

m1 = 0 m2 = 0

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 1 2 1 2 1 1 2 1 1 2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 1 2 1 2 1 1 2 1 1 2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

If all slopes are 0 (“flat” coloring) we get an HHF on Td

n.

Td

n

Zd

SLIDE 35

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Flat Slope HHFs ↔ HHFs on Td

n

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 1 2 1 2 1 1 2 1 1 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

m1 = 0 m2 = 0

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 1 2 1 2 1 1 2 1 1 2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 1 2 1 2 1 1 2 1 1 2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

Flat 3-colorings

HHFs Iff all slopes are 0 (“flat” coloring) we get an HHF on Td

n.

Td

n

Zd

SLIDE 36

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Pulling the HHFs result to 3-colorings

3-colorings of Td

n

Periodic 3-colorings of Zd

2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 1 1 2 1 2 1 1 1 1 2 2 2 1 1 2 1 2

SLIDE 37

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Pulling the HHFs result to 3-colorings

3-colorings of Td

n

Periodic 3-colorings of Zd
Quasi-periodic HHFs on Zd

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

SLIDE 38

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Pulling the HHFs result to 3-colorings

3-colorings of Td

n

Periodic 3-colorings of Zd
Quasi-periodic HHFs on Zd

Periodic HHFs on Zd

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

SLIDE 39

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Pulling the HHFs result to 3-colorings

3-colorings of Td

n

Periodic 3-colorings of Zd
Quasi-periodic HHFs on Zd

Periodic HHFs on Zd

HHFs on Td

n

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

SLIDE 40

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Pulling the HHFs result to 3-colorings

3-colorings of Td

n

Periodic 3-colorings of Zd
Quasi-periodic HHFs on Zd

Periodic HHFs on Zd

HHFs on Td

n

(which are known to be rigid)

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

SLIDE 41

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Pulling the HHFs result to 3-colorings

3-colorings of Td

n

Periodic 3-colorings of Zd
Quasi-periodic HHFs on Zd

≈ Periodic HHFs on Zd

HHFs on Td

n

(which are known to be rigid) GOAL: Show that most quasi-periodic HHFs are periodic.

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

SLIDE 42

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Proving Most Quasi-periodic are Periodic

We construct a “flattening” map Ψ from quasi-periodic HHFs into periodic ones.

1:1 Few:1

(mod3)

1

Ψ

HHFs on Td

n

small subset “sloped” 3-colorings “flat” 3-colorings

SLIDE 43

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Flattening the slope

Introducing the reflection Ψ

Denote QPm := {h ∈ QP : m is the slope of h} We construct Ψm : QPm → QP0, a one-to-one mapping.

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

Ψm →

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

SLIDE 44

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Flattening the slope

Introducing the reflection Ψ

Denote QPm := {h ∈ QP : m is the slope of h} We construct Ψm : QPm → QP0, a one-to-one mapping.

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

Ψm →

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 1

Observe that the image contains a long level set.

SLIDE 45

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Flattening the slope

Introducing the reflection Ψ

Denote QPm := {h ∈ QP : m is the slope of h} We construct Ψm : QPm → QP0, a one-to-one mapping.

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

Ψm →

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 1

Observe that the image contains a long level set. Peled 2010: Long level sets are extremely uncommon.

SLIDE 46

Introduction Proof overview Ideas and Method Open problems Homomorphism height functions 3-Coloring Td

n = Quasi Periodic HHF on Zd

Quasi-periodic to periodic

Flattening the slope

Introducing the reflection Ψ

Denote QPm := {h ∈ QP : m is the slope of h} We construct Ψm : QPm → QP0, a one-to-one mapping.

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

Ψm →

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 1

Observe that the image contains a long level set. Peled 2010: Long level sets are extremely uncommon. We deduce the image of Ψm is small.

SLIDE 47

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Ideas and Method

SLIDE 48

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening Intuition

One-dimensional intuition: reflection.

1 2 3 4 5 6 1 2 1 1 2 3 4 5 6 1 2 1

SLIDE 49

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening Intuition

One-dimensional intuition: reflection. Where to reflect?

1 2 3 4 5 6 1 2 1 1 2 3 4 5 6 1 2 1

SLIDE 50

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening Intuition

One-dimensional intuition: reflection. Where to reflect?

immediately after height m1

2

1 2 3 4 5 6 1 2 1 1 2 3 4 5 6 1 2 1

SLIDE 51

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening Intuition

One-dimensional intuition: reflection. Where to reflect?

immediately after height m1

2

Problem: several mi-s. Can we fix them all at once?

1 2 3 4 5 6 1 2 1 1 2 3 4 5 6 1 2 1

SLIDE 52

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening Intuition

One-dimensional intuition: reflection. Where to reflect?

immediately after height m1

2

Problem: several mi-s. Can we fix them all at once? Answer: Topology says - Yes.

1 2 3 4 5 6 1 2 1 1 2 3 4 5 6 1 2 1

SLIDE 53

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening multi-dimensional functions

Quasi-periodic functions are homotopy equivalent to linear ones.

3 9 3 6 3 12 6 6 9 12 15 15 12 15 18 21 24 6 15 3 15 27 18 3

SLIDE 54

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening multi-dimensional functions

Quasi-periodic functions are homotopy equivalent to linear ones.

3 9 3 6 3 12 6 6 9 12 15 15 12 15 18 21 24 6 15 3 15 27 18 3

Two types of level contours: Trivial level contours.

SLIDE 55

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening multi-dimensional functions

Quasi-periodic functions are homotopy equivalent to linear ones.

3 9 3 6 3 12 6 6 9 12 15 15 12 15 18 21 24 6 15 3 15 27 18 3

Two types of level contours: Trivial level contours. Non-Trivial level contours.

SLIDE 56

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening multi-dimensional functions

Quasi-periodic functions are homotopy equivalent to linear ones.

3 9 3 6 3 12 6 6 9 12 15 15 12 15 18 21 24 6 15 3 15 27 18 3

We pick particular non-trivial level contours.

SLIDE 57

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening multi-dimensional functions

Quasi-periodic functions are homotopy equivalent to linear ones.

3 9 3 6 3 12 6 6 9 12 15 15 12 15 18 21 24 6 15 3 15 27 18 3

We pick particular non-trivial level contours. We find the proper reflection “domain” on the torus.

SLIDE 58

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Flattening multi-dimensional functions

Quasi-periodic functions are homotopy equivalent to linear ones.

3

3

3 3

3

3 3 3 0 -3 3 3 3 3 3

We pick particular non-trivial level contours. We find the proper reflection “domain” on the torus. We make the reflection.

SLIDE 59

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

How to discretize?

Challenges of the discrete setting:

Define level sets properly. Establish their structure. Identify trivial level sets. Prove the invertability of the reflection.

SLIDE 60

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

How to discretize?

Challenges of the discrete setting:

Define level sets properly. Establish their structure. Identify trivial level sets. Prove the invertability of the reflection. We will focus on these in this presentation.

SLIDE 61

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Sublevel sets

Towards level sets

Sublevel set of v at height k LCk

h(v) is the connected component of v in G \{u ∈ G | h(u) = k}

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2

1

2 2 2 2

1

2 2 2 2

1
1
1

3

SLIDE 62

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Sublevel Components

The fundament of level sets

Sublevel component from v to u at height k LCk

h(v, u) is the complement of the connected component of u in

G \ LCk

h(v)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2

1
1
1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3

LCk

h(v)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2

1
1
1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3

LCk

h(v, u)

SLIDE 63

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Sublevel Components

The fundament of level sets

Sublevel component from v to u at height k LCk

h(v, u) is the complement of the connected component of u in

G \ LCk

h(v)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2

1
1
1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2 2 2 2 2 2 2 3 2 2 1 1 1 1 1 2 2 2 2 2 2 2

LCk

h(v)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2

1
1
1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3

LCk

h(v, u)

SLIDE 64

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Sublevel Components

The fundament of level sets

Sublevel component from v to u at height k LCk

h(v, u) is the complement of the connected component of u in

G \ LCk

h(v)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2

1
1
1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2 2 2 2 2 2 2 3 2 2 1 1 1 1 1 2 2 2 2 2 2 2

LCk

h(v)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

1

2 2 2

1
1
1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3

LCk

h(v, u)

The edge boundary of a sublevel component is called a level set.

SLIDE 65

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

3 Types of Level Components

A trichotomy

For t ∈ nZd, and a set U ⊂ Zd we call U + t a translate of U.

SLIDE 66

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

3 Types of Level Components

A trichotomy

For t ∈ nZd, and a set U ⊂ Zd we call U + t a translate of U. 3 types of level components Let U = LCk

h(u, v) be a sublevel component with non-empty

boundary. One of the following holds:

(Trivial) All of U’s translates are disjoint. (Trivial) All of Uc’s translates are disjoint. (Non-trivial) The translates of U are totally ordered by inclusion.

SLIDE 67

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Trichotomy - example

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

Example

SLIDE 68

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Trichotomy - example

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1
2
2
1
1
1
1
1
1

Trivial (Disjoint translates)

SLIDE 69

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Trichotomy - example

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 2 3 2 3 3 3 2 2 2 2 2 3 1 1 1 1

1

1 1 1

1
1

2 2 2 3 2 3 3 3 2 2 2 2 2 3 1 1 1 1

1

1 1 1

1
1

3 1 2 2 1 1 2 3 3 1 2 2 1 1 4 2 3 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 3 2 3 3 3 3 3 2 2 1 1 1 1 1 0 -1 0 2 2 3 3 3 2 2 2 1 1

1

1 1 1 1 1 1 1

1

1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 3 2 3 3 3 3 3 2 2 1 1 1 1 1 0 -1 0 1 1 1 1 1 1 1

1

2 2 3 3 3 2 2 2 1 1

1

1 1

1
1

1

1
1
2
2
1
1
1
1
1
1
2
2
1
1
1
1
1
1

Trivial (Disjoint complement translates)

SLIDE 70

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Trichotomy - example

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 2 2 2 2 2 2 2 2 3 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1 1 1 1 1 1 1 1 1 0 -1

1
1

2 2 3 3 3 2 2 2 1 1

1

2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 2 2 2 2 2 2 2 2 3 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1 1 1 1 1 1 1 1 1 0 -1

1

1 1 1 1 1 1 1 1 1 1 1 1

1
1
1

2 2 3 3 3 2 2 2 1 1

1

1 1

1
1

1 1 1 1 1 1 1 1 1 1 1 1

1
1
1

1 1

1
1

4

1 -2
2
1
1
1
1
1
2
2
1
1
1
1
1

Non-Trivial (Ordered)

SLIDE 71

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Trichotomy - example

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

Trivial sublevel components do not create slope.

SLIDE 72

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Formula for heights

Denote L = {A : ∃u1, u2 ∈ Zd : A = LC h(u2)

h

(u1, u2)} Formula for h(u) − h(v) h(u) − h(v) = |{A ∈ L : v ∈ A, u / ∈ A}| − |{A ∈ L : v / ∈ A, u ∈ A}|

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

SLIDE 73

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Formula for heights

Denote L = {A : ∃u1, u2 ∈ Zd : A = LC h(u2)

h

(u1, u2)} Formula for h(u) − h(v) h(u) − h(v) = |{A ∈ L : v ∈ A, u / ∈ A}| − |{A ∈ L : v / ∈ A, u ∈ A}| Specializing to v = u + t for t ∈ nZd we write: Formula for h(u) − h(v) h(u)−h(v) = |{A ∈ L′ : v ∈ A, u / ∈ A}|−|{A ∈ L′ : v / ∈ A, u ∈ A}| where L′ is the set of non-trivial sublevel components in L.

SLIDE 74

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

Formula for heights

Denote L = {A : ∃u1, u2 ∈ Zd : A = LC h(u2)

h

(u1, u2)} Formula for h(u) − h(v) h(u) − h(v) = |{A ∈ L : v ∈ A, u / ∈ A}| − |{A ∈ L : v / ∈ A, u ∈ A}| Specializing to v = u + t for t ∈ nZd we write: Formula for h(u) − h(v) h(u)−h(v) = |{A ∈ L′ : v ∈ A, u / ∈ A}|−|{A ∈ L′ : v / ∈ A, u ∈ A}| where L′ is the set of non-trivial sublevel components in L. Used, for example, to show the existence of non-trivial sublevel components for sloped function.

SLIDE 75

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

The Discrete Picture

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

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13

SLIDE 76

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

The Discrete Picture

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

SLIDE 77

Introduction Proof overview Ideas and Method Open problems Flattening the slope From continuous topology to a discrete one Application of the trichotomy

The Discrete Picture

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

4 2 2 2 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2
2

2 2

1

2 2 2 2 2 2 3 1 2 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 1

1

1 1 1 1 1 1 1 1 0 -1

1
1
1
1
1
1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

2 2 3 1 1 3 3 2 2 2 1 1

1
1
1

SLIDE 78

Introduction Proof overview Ideas and Method Open problems Open problems

Open Problems

Ordered by estimated difficulty

Odd Tori.

SLIDE 79

Introduction Proof overview Ideas and Method Open problems Open problems

Open Problems

Ordered by estimated difficulty

Odd Tori. 4-colors and more.

SLIDE 80

Introduction Proof overview Ideas and Method Open problems Open problems

Open Problems

Ordered by estimated difficulty

Odd Tori. 4-colors and more. Non-zero temperature.

SLIDE 81

Introduction Proof overview Ideas and Method Open problems Open problems

Open Problems

Ordered by estimated difficulty

Odd Tori. 4-colors and more. Non-zero temperature. Low dimension.

SLIDE 82

Introduction Proof overview Ideas and Method Open problems Open problems