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
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
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
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
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
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
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
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)
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 6 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18
The perceptron
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 7 / 18
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
The perceptron – persistence homology
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 9 / 18
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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)
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)
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
The perceptron – persistence homology
Y. Kallus (Santa Fe Institute) Configuration space topology ICERM, Oct. 17, 2016 19 / 18