Computational topology of con guration spaces - ICERM · Computational topology of con guration...

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Computational topology of configuration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits ICERM, Providence October 17, 2016 Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 1 / 18

Transcript of Computational topology of con guration spaces - ICERM · Computational topology of con guration...

Page 1: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Computational topology ofconfiguration spaces

Yoav Kallus

Santa Fe Institute

Stochastic Topology and Thermodynamic LimitsICERM, ProvidenceOctober 17, 2016

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 1 / 18

Page 2: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Clustering of the phase space

φVolume fraction,

Pre

ssure

, P

φ φ φth K GCPd

φ

Σ

j

φ

Σequilibrium

snon−planted−clusters

planted−clusters

stot

CsCd CcAverage degree

14

13.6 13.4 13.2 13 12.8 12.6

0.2

0.15

0.1

0.05

0 13.8

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 2 / 18

Page 3: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Clustering of the phase space

φVolume fraction,

Pre

ssure

, P

φ φ φth K GCPd

φ

Σ

j

φ

Σequilibrium

snon−planted−clusters

planted−clusters

stot

CsCd CcAverage degree

14

13.6 13.4 13.2 13 12.8 12.6

0.2

0.15

0.1

0.05

0 13.8

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 2 / 18

Page 4: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Clustering of the phase space

φVolume fraction,

Pre

ssure

, P

φ φ φth K GCPd

φ

Σ

j

φ

Σequilibrium

snon−planted−clusters

planted−clusters

stot

CsCd CcAverage degree

14

13.6 13.4 13.2 13 12.8 12.6

0.2

0.15

0.1

0.05

0 13.8

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 2 / 18

Page 5: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

k-SAT clustering

Theoremβ, γ, θ, δ and εk → 0 exist such that for a random k-SATformula with n variables and m = αn clauses, where

(1 + εk)2k log(k)/k ≤ α ≤ (1− εk)2k log(2)

the solution can is partitioned w.h.p. into clusters, s.t.

there are ≥ exp(βn) clusters,

any cluster has ≤ exp(−γn) of all solutions,

solutions in distinct clusters are ≥ δn apart, and

any connecting path violates ≥ θn clauses along it.

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 3 / 18

Moore & Mertens, The Nature of Computation

Page 6: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Clustering phenomenology

All about connected components of the configurationspace. What about higher dimensional topologicalinvariants?

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 4 / 18

Moore & Mertens, The Nature of Computation

Page 7: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Clustering phenomenology

All about connected components of the configurationspace. What about higher dimensional topologicalinvariants?

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 4 / 18

Moore & Mertens, The Nature of Computation

Page 8: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

5 disks in a square

0.1000

0.1464 0.1306 0.12500.1667

0.1686 0.1686 0.14790.16020.1667 0.16670.1681

0.17050.1942 0.16920.1871 0.1693

0.2071 0.1964 0.1705

1

2

3

4

5

6

2

2

2

3

3

3

4

4 4

4 4

4 4

5

5 5

6

6

6 6

6

6 6

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 5 / 18

Carlsson, Gorham, Kahle, & Mason, Phys. Rev. E 85, 011303 (2012)

Page 9: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18

Page 10: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18

Page 11: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18

Page 12: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18

Page 13: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18

Page 14: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18

Page 15: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18

Page 16: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18

Page 17: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18

Page 18: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18

Page 19: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron – Betti numbers

0

5

10

15

20

25

n = 6

0

10

20

30

40

50

60

70

n = 7

0.0 0.2 0.4 0.6 0.8 1.01/R^2

0

50

100

150

200

n = 8

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 8 / 18

Page 20: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron – persistence homology

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 9 / 18

Page 21: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Stochastic topology, a different criticality

d fixed, n→∞: structure only at local scaled , n→∞, d ∼ exp(n2): no spatial structured , n→∞, d ∼ n: structure at all scales

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 10 / 18

Page 22: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Stochastic topology, a different criticality

d fixed, n→∞: structure only at local scaled , n→∞, d ∼ exp(n2): no spatial structured , n→∞, d ∼ n: structure at all scales

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 10 / 18

Page 23: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The sphere packing problem

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 11 / 18

log2 δ + 196n(24 − n)

Conway & Sloane, SPLAG

Page 24: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Packing problem restricted to lattices

Restricted to lattices, what is the densest packingstructure?

n L2 A2 Lagrange (1773)3 D3 = A3 Gauss (1840)4 D4 Korkin & Zolotarev (1877)5 D5 Korkin & Zolotarev (1877)6 E6 Blichfeldt (1935)7 E7 Blichfeldt (1935)8 E8 Blichfeldt (1935)

24 Λ24 Cohn & Kumar (2004)

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 12 / 18

Page 25: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The space of latticesL = AZn

, so L =

O(n)\

GLn(R)

/GLn(Z)

But AZn and RAZn are isometric if R is a rotation.But AZn and AQZn are the same lattice if QZn = Zn.

O(n)\GLn(R) = Sn>0 ⊂ Symn, the space of symmetric,positive definite matrices: take G = ATA.

Symn

Sn>0

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 13 / 18

Page 26: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The space of latticesL = AZn, so L =

O(n)\

GLn(R)

/GLn(Z)But AZn and RAZn are isometric if R is a rotation.But AZn and AQZn are the same lattice if QZn = Zn.

O(n)\GLn(R) = Sn>0 ⊂ Symn, the space of symmetric,positive definite matrices: take G = ATA.

Symn

Sn>0

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 13 / 18

Page 27: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The space of latticesL = AZn, so L =

O(n)\

GLn(R)

/GLn(Z)

But AZn and RAZn are isometric if R is a rotation.

But AZn and AQZn are the same lattice if QZn = Zn.

O(n)\GLn(R) = Sn>0 ⊂ Symn, the space of symmetric,positive definite matrices: take G = ATA.

Symn

Sn>0

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 13 / 18

Page 28: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The space of latticesL = AZn, so L =O(n)\GLn(R)

/GLn(Z)

But AZn and RAZn are isometric if R is a rotation.

But AZn and AQZn are the same lattice if QZn = Zn.

O(n)\GLn(R) = Sn>0 ⊂ Symn, the space of symmetric,positive definite matrices: take G = ATA.

Symn

Sn>0

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 13 / 18

Page 29: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The space of latticesL = AZn, so L =O(n)\GLn(R)/GLn(Z)But AZn and RAZn are isometric if R is a rotation.But AZn and AQZn are the same lattice if QZn = Zn.

O(n)\GLn(R) = Sn>0 ⊂ Symn, the space of symmetric,positive definite matrices: take G = ATA.

Symn

Sn>0

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 13 / 18

Page 30: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The space of latticesL = AZn, so L =O(n)\GLn(R)/GLn(Z)But AZn and RAZn are isometric if R is a rotation.But AZn and AQZn are the same lattice if QZn = Zn.

O(n)\GLn(R) = Sn>0 ⊂ Symn, the space of symmetric,positive definite matrices: take G = ATA.

Symn

Sn>0

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 13 / 18

Page 31: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The Ryshkov polyhedronWe are interested in the lattices with packing radius ≥ 1:{G ∈ Sn>0 : n · Gn ≥ 1 for all n ∈ Zn}.

n1 ·Gn1 ≥ 0

Sn>0

Symn

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 14 / 18

Page 32: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The Ryshkov polyhedronWe are interested in the lattices with packing radius ≥ 1:{G ∈ Sn>0 : n · Gn ≥ 1 for all n ∈ Zn}.

Symn

Sn>0

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 14 / 18

Page 33: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The Ryshkov polyhedronWe are interested in the lattices with packing radius ≥ 1:{G ∈ Sn>0 : n · Gn ≥ 1 for all n ∈ Zn}.

Symn

Sn>0

R

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 14 / 18

Page 34: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The Ryshkov polyhedronThe polytope is locally finite, and has finitely many facesmodulo GLn(Z) action

Symn

Sn>0

R

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 14 / 18

Page 35: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The Ryshkov polyhedronWe determinant (equivalently, density) gives a filtrationof the space

Symn

Sn>0

R

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 14 / 18

Page 36: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Lattice RCP

2 5 10 20 50

101

102

103

slope = 1

slope = 2.61

d

2dϕ

0.85 0.9 0.95 1 1.05 1.1 1.15

0

10

20

30

0

ϕ/〈ϕ〉

〈ϕ〉P

(ϕ)

15161718192021222324

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 15 / 18

K, Marcotte, & Torquato, Phys. Rev. E 88, 062151 (2013)

Page 37: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Pair correlations and quasicontacts

1 1.2 1.4 1.6 1.8

0.9

1

1.1

1

r

g(r)

15 16 1718 19 2021 22 2324

10−4 10−3

10−1

100

r − 1(r

−1)(dZ

/dr)

15 16 1718 19 2021 22 2324

g(r) ∼ (r − 1)−γ

Z (r) ∼ d(d + 1) + Ad(r − 1)1−γ

γ = 0.314± 0.004

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 16 / 18

K, Marcotte, & Torquato, Phys. Rev. E 88, 062151 (2013)

Page 38: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Contact force distribution

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

f/〈f〉

P(f

/〈f

〉)

15161718192021222324

2.5 3 3.5 410−4

10−3

10−2

10−1

10−3 10−2 10−1

10−5

10−4

10−3

10−2

10−1

f/〈f〉(f

/〈f

〉)P(f

/〈f

〉)

P(f ) ∼ f θ

θ = 0.371± 0.010

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 17 / 18

K, Marcotte, & Torquato, Phys. Rev. E 88, 062151 (2013)

Page 39: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

Topology of the space of lattices

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 18 / 18

Elbaz-Vincent, Gangl, & Soule, Perfect forms and the cohomology of modulargroups, arXiv:1001.0789

Page 40: Computational topology of con guration spaces - ICERM · Computational topology of con guration spaces Yoav Kallus Santa Fe Institute Stochastic Topology and Thermodynamic Limits

The perceptron – persistence homology

Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 19 / 18