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