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,

φ φ φth K GCPd

Σequilibrium

snon−planted−clusters

planted−clusters

CsCd CcAverage degree

13.6 13.4 13.2 13 12.8 12.6

0 13.8

φVolume fraction,

φ φ φth K GCPd

Σequilibrium

planted−clusters

13.6 13.4 13.2 13 12.8 12.6

0 13.8

φVolume fraction,

φ φ φth K GCPd

Σequilibrium

planted−clusters

13.6 13.4 13.2 13 12.8 12.6

0 13.8

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.

Moore & Mertens, The Nature of Computation

Clustering phenomenology

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

Clustering phenomenology

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

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

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

The perceptron

The perceptron – Betti numbers

0.0 0.2 0.4 0.6 0.8 1.01/R^2

The perceptron – persistence homology

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

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

The sphere packing problem

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)

The space of latticesL = AZn

, so L =

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.

The space of latticesL = AZn, so L =

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.

The space of latticesL = AZn, so L =

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.

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.

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.

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

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

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

Lattice RCP

2 5 10 20 50

slope = 1

slope = 2.61

0.85 0.9 0.95 1 1.05 1.1 1.15

ϕ/〈ϕ〉

〈ϕ〉P

15161718192021222324

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

Pair correlations and quasicontacts

1 1.2 1.4 1.6 1.8

15 16 1718 19 2021 22 2324

10−4 10−3

10−1

r − 1(r

−1)(dZ

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

Contact force distribution

0 0.5 1 1.5 2 2.5 3 3.50

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

〉)P(f

P(f ) ∼ f θ

θ = 0.371± 0.010

Topology of the space of lattices

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

The perceptron – persistence homology

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

Documents

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

Lecture notes on Topology

1 Using Local Geometry for Tunable Topology Control in ...anrg.usc.edu/www/papers/PoduriPattemKrishnamachariSukhatme_TM… · Using Local Geometry for Tunable Topology Control in

Tensor topology - homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Enriquemoliner.pdf · Tensor topology Pau Enrique Moliner1 Chris Heunen 1 Sean Tull 2 1University

Topology of the space of D-minimal metrics

Computational Simulations

Symmetry, Topology and Phases of Matter · 2016-05-21 · Topology and Band Theory I I. Introduction - Insulating state, topology and band theory II. Band Topology in One Dimension

Computational Semantics: More Calculuscourses.washington.edu › ling571 › ling571_fall_2010 › slides › comps… · Computational Semantics: More Calculus Scott Farrar CLMA,

MIE SCATTERING COMPUTATIONAL ELECTROMAGNETICS …

Rainer Herges- Topology in Chemistry: Designing Mobius Molecules

Topology of “ white ” stars in relativistic fragmentation of

ΓΕ77 COMPUTATIONAL LINGUISTICS

Computational Electromagnetics : Antenna computations

Turkish Computational and Theoretical Chemistry

Multi-Objective Topology Optimization...structural analysis tools. The increase in computational power have also lead to the development of advanced computer aided design (CAD) softwares.

Computational+investigation+intothe+gas1phase+ozonolysis+ … · Computational 26! characterisationofthepotential!energysurface !thusprovidesinsightintothis! 27! previouslyunstudied!system,andwillaid

Supplementary Figure 1: NOON State Device Con guration ......Supplementary Figure 1: NOON State Device Con guration. This microscope image of the NOON state device shows how the Mach-Zehnder

Geometric Topology and Geometry of Banach Spacesarkady/Eilat_2017_conference/booklet... · Geometric Topology and Geometry of Banach Spaces ... Variants of decomposition complexity

Intermolecular Michael Reactions: A Computational ...evans.rc.fas.harvard.edu/pdf/evans330.pdfIntermolecular Michael Reactions: A Computational Investigation ... P. Conjugate Additions

Computational Methods in Population Genetics

Computational Fluid Dynamics · PDF fileSeite 1Introduction j Computational Fluid Dynamics j 01.02.2017 Computational Fluid Dynamics Theory, Numerics, Modelling Lucas Engelhardt SISO