Random topology and geometry - Statistical Sciencesayan/JMM/randomtopology.pdf · topological...

Random topology and geometry

Matthew Kahle

Ohio State University

AMS Short Course

Matthew Kahle (Ohio State University) Random topology and geometry AMS Short Course

Introduction


“I predict a new subject of statistical topology. Rather than count thenumber of holes, Betti numbers, etc., one will be more interested in thedistribution of such objects on noncompact manifolds as one goes outto infinity.” — Isadore Singer, 2004.


Randomness models the natural world.

We need probability in order to quantify the significance oftopological “features” detected with topological data analysis.Certain situations in physics seem to be well modeled by randomtopology: cosmology, black holes, quantum gravity, etc.Can we explain why so many groups / manifolds / simplicialcomplexes / etc. seem to have a certain topological property?

E.g. many simplicial complexes and posets arising incombinatorics are homotopy equivalent to bouquets of spheres.But why does this happen so often?



We need probability in order to quantify the significance oftopological “features” detected with topological data analysis.

Certain situations in physics seem to be well modeled by randomtopology: cosmology, black holes, quantum gravity, etc.Can we explain why so many groups / manifolds / simplicialcomplexes / etc. seem to have a certain topological property?




We need probability in order to quantify the significance oftopological “features” detected with topological data analysis.Certain situations in physics seem to be well modeled by randomtopology: cosmology, black holes, quantum gravity, etc.

Can we explain why so many groups / manifolds / simplicialcomplexes / etc. seem to have a certain topological property?











The probabilistic method provides existence proofs.

Ramsey theory and extremal graph theory e.g. ErdosGeometric group theory — e.g. Gromov, ZukMetric geometry — e.g. BourgainExpander graphs — e.g. Pinsker, Barzdin & Kolmogorov



Ramsey theory and extremal graph theory e.g. Erdos

Geometric group theory — e.g. Gromov, ZukMetric geometry — e.g. BourgainExpander graphs — e.g. Pinsker, Barzdin & Kolmogorov



Ramsey theory and extremal graph theory e.g. ErdosGeometric group theory — e.g. Gromov, Zuk

Metric geometry — e.g. BourgainExpander graphs — e.g. Pinsker, Barzdin & Kolmogorov



Ramsey theory and extremal graph theory e.g. ErdosGeometric group theory — e.g. Gromov, ZukMetric geometry — e.g. Bourgain

Expander graphs — e.g. Pinsker, Barzdin & Kolmogorov



Ramsey theory and extremal graph theory e.g. ErdosGeometric group theory — e.g. Gromov, ZukMetric geometry — e.g. BourgainExpander graphs — e.g. Pinsker, Barzdin & Kolmogorov



Some earlier work:

Early mention by Milnor — 1964Gaussian fields on manifolds — Adler and Taylor, 2003.Random triangulated surfaces — Pippenger and Schleich, 2006.Random 3-manifolds — Dunfield and Thurston, 2006.Random planar linkages — Farber and Kappeller, 2007.Random simplicial complexes — Linial and Meshulam, 2006.


Some earlier work:

Early mention by Milnor — 1964

Gaussian fields on manifolds — Adler and Taylor, 2003.Random triangulated surfaces — Pippenger and Schleich, 2006.Random 3-manifolds — Dunfield and Thurston, 2006.Random planar linkages — Farber and Kappeller, 2007.Random simplicial complexes — Linial and Meshulam, 2006.


Some earlier work:

Early mention by Milnor — 1964Gaussian fields on manifolds — Adler and Taylor, 2003.

Random triangulated surfaces — Pippenger and Schleich, 2006.Random 3-manifolds — Dunfield and Thurston, 2006.Random planar linkages — Farber and Kappeller, 2007.Random simplicial complexes — Linial and Meshulam, 2006.


Some earlier work:

Early mention by Milnor — 1964Gaussian fields on manifolds — Adler and Taylor, 2003.Random triangulated surfaces — Pippenger and Schleich, 2006.

Random 3-manifolds — Dunfield and Thurston, 2006.Random planar linkages — Farber and Kappeller, 2007.Random simplicial complexes — Linial and Meshulam, 2006.


Some earlier work:

Early mention by Milnor — 1964Gaussian fields on manifolds — Adler and Taylor, 2003.Random triangulated surfaces — Pippenger and Schleich, 2006.Random 3-manifolds — Dunfield and Thurston, 2006.

Random planar linkages — Farber and Kappeller, 2007.Random simplicial complexes — Linial and Meshulam, 2006.


Some earlier work:

Early mention by Milnor — 1964Gaussian fields on manifolds — Adler and Taylor, 2003.Random triangulated surfaces — Pippenger and Schleich, 2006.Random 3-manifolds — Dunfield and Thurston, 2006.Random planar linkages — Farber and Kappeller, 2007.

Random simplicial complexes — Linial and Meshulam, 2006.


Some earlier work:

Early mention by Milnor — 1964Gaussian fields on manifolds — Adler and Taylor, 2003.Random triangulated surfaces — Pippenger and Schleich, 2006.Random 3-manifolds — Dunfield and Thurston, 2006.Random planar linkages — Farber and Kappeller, 2007.Random simplicial complexes — Linial and Meshulam, 2006.


Random graphs


Define G(n,p) to be the probability space of graphs on vertex set[n] = {1,2, . . . ,n}, where each edge has probability p, independently.

We use the notation G ∼ G(n,p) to indicate a graph chosen accordingto this distribution.

It is often useful to think of a stochastic process associated withG(n,p) where edges are added one at a time.































Usually p is a function of n and n→∞.

We are often interested in thresholds for graph properties. Anarchetypal example is the Erdos–Rényi theorem.


Usually p is a function of n and n→∞.

We are often interested in thresholds for graph properties. Anarchetypal example is the Erdos–Rényi theorem.


Theorem (Erdos–Rényi, 1959)Let ε > 0 be fixed and G ∼ G(n,p). Then

P[G is connected]→

1 : p ≥ (1 + ε) log n/n

0 : p ≤ (1− ε) log n/n

They actually proved a slightly sharper result.


Theorem (Erdos–Rényi, 1959)Let ε > 0 be fixed and G ∼ G(n,p). Then

P[G is connected]→

1 : p ≥ (1 + ε) log n/n

0 : p ≤ (1− ε) log n/n

They actually proved a slightly sharper result.


Theorem (Erdos–Rényi, 1959)Let c ∈ R be fixed and G ∼ G(n,p). If

p =log n + c

n,

then β0(G) is asymptotically Poisson distributed with mean e−c and inparticular

P[G is connected]→ e−e−c

as n→∞.


The first step is to show that if p ≈ log n/n, then the probability thatthere are any components of order i , with 2 ≤ i ≤ n/2 tends to 0 asn→∞.

A union bound argument shows that it is sufficient to show that

bn/2c∑k=2

(nk

)kk−2pk−1(1− p)k(n−k) → 0,

as n→∞.

So if p ≈ log n/n, then w.h.p. G(n,p) consists of a giant componentand isolated vertices.




bn/2c∑k=2

(nk

)kk−2pk−1(1− p)k(n−k) → 0,

as n→∞.





bn/2c∑k=2

(nk

)kk−2pk−1(1− p)k(n−k) → 0,

as n→∞.



Now set p = (log n + c)/n, where c ∈ R is fixed. By linearity ofexpectation, the expected number of isolated vertices V is

E[V ] = n(1− p)n−1,

and since 1− p ≈ e−p for p ≈ 0, we have

E[V ]→ e−c ,

as n→∞.

By computing the higher moments, or by Stein’s method, one canshow that V approaches a Poisson distribution with mean e−c .


Now set p = (log n + c)/n, where c ∈ R is fixed. By linearity ofexpectation, the expected number of isolated vertices V is

E[V ] = n(1− p)n−1,

and since 1− p ≈ e−p for p ≈ 0, we have

E[V ]→ e−c ,

as n→∞.

By computing the higher moments, or by Stein’s method, one canshow that V approaches a Poisson distribution with mean e−c .


Comments:

The proof is phrased in terms of cohomology rather than in termsof homology.The ultimate obstruction to connectivity is isolated vertices. Atheorem of Bollobás and Thomason takes this idea to its naturalconclusion.At the connectivity threshold, G ∼ G(n,p) is already an expander.


Comments:The proof is phrased in terms of cohomology rather than in termsof homology.

The ultimate obstruction to connectivity is isolated vertices. Atheorem of Bollobás and Thomason takes this idea to its naturalconclusion.At the connectivity threshold, G ∼ G(n,p) is already an expander.


Comments:The proof is phrased in terms of cohomology rather than in termsof homology.The ultimate obstruction to connectivity is isolated vertices. Atheorem of Bollobás and Thomason takes this idea to its naturalconclusion.

At the connectivity threshold, G ∼ G(n,p) is already an expander.


Comments:The proof is phrased in terms of cohomology rather than in termsof homology.The ultimate obstruction to connectivity is isolated vertices. Atheorem of Bollobás and Thomason takes this idea to its naturalconclusion.At the connectivity threshold, G ∼ G(n,p) is already an expander.


The normalized graph Laplacian L of a connected graph H is thematrix

L = I − D−1/2AD−1/2,

where I is the identity matrix, D is the diagonal matrix with degreesdown the diagonal, and A is the adjacency matrix.

The eigenvalues satisfy 0 = λ1 < λ2 ≤ . . . λn ≤ 2, and λ2 is oftencalled the spectral gap of the graph.

If {Hi} is a sequence of graphs such that #V (Hi)→∞ and such that

lim infλ2[Hi ] > 0,

as i →∞, we say that {Hi} is an expander family.



L = I − D−1/2AD−1/2,








L = I − D−1/2AD−1/2,







Theorem (Hoffman, K., Paquette, 2013)Fix k ≥ 0 and let ω →∞ arbitrarily slowly. Let0 = λ1 ≤ λ2 ≤ · · · ≤ λn ≤ 2 be the eigenvalues of the normalizedgraph Laplacian of G ∼ G(n,p). If

p ≥ (k + 1) log n + ω

n,

then

1−

√Cnp≤ λ2 ≤ · · · ≤ λn ≤ 1 +

√Cnp,

with probability at least 1− o(n−k).


Random simplicial complexes


Linial and Meshulam defined Y2(n,p) to be the probability space of2-dimensional simplicial complexes with vertex set [n], edge set

([n]2

),

and such that each 2-face has probability p.


More generally, Meshulam and Wallach defined the randomd-dimensional simplicial complex Yd (n,p) to be the probabilitydistribution over all simplicial complexes on n vertices with completed − 1-skeleton, and such that every d-dimensional face is includedwith probability p, independently.












The following theorems generalize the Erdos–Rényi theorem to higherdimensions, and inspired a lot of future work by various teams ofresearchers.

Theorem(Linial–Meshulam, 2006) Let ε > 0 be fixed and Y ∼ Y2(n,p). Then

P[H1(Y ,Z/2) = 0]→

1 : p ≥ (2+ε) log n

n

0 : p ≤ (2−ε) log nn


The following theorems generalize the Erdos–Rényi theorem to higherdimensions, and inspired a lot of future work by various teams ofresearchers.

Theorem(Linial–Meshulam, 2006) Let ε > 0 be fixed and Y ∼ Y2(n,p). Then

P[H1(Y ,Z/2) = 0]→

1 : p ≥ (2+ε) log n

n

0 : p ≤ (2−ε) log nn


Theorem(Meshulam–Wallach, 2008) Let d ≥ 2, ` ≥ 2, and ε > 0 be fixed andY ∼ Yd (n,p). Then

P[Hd−1(Y ,Z/`) = 0]→

1 : p ≥ (d+ε) log n

n

0 : p ≤ (d−ε) log nn


These theorems rely on deep combinatorics, and in particular careful“cocycle counting” arguments.


The threshold for simple connectivity is much larger.

Theorem(Babson–Hoffman–K., 2011) Let ε > 0 be fixed and Y ∼ Y2(n,p). Then

P[π1(Y ) = 0]→

1 : p ≥ nε

√n

0 : p ≤ n−ε√

n


The threshold for simple connectivity is much larger.

Theorem(Babson–Hoffman–K., 2011) Let ε > 0 be fixed and Y ∼ Y2(n,p). Then

P[π1(Y ) = 0]→

1 : p ≥ nε

√n

0 : p ≤ n−ε√

n


We recently revisited homology vanishing theorems for randomd-dimensional complexes.


Besides the spectral gap theorem, our main tool is the following.

Theorem (Garland, Ballman–Swiatkowski)If ∆ is a finite, pure d-dimensional, simplicial complex, such that

λ2[lk(σ)] > 1− 1d

for every (d − 2)-dimensional face σ, then Hd−1(∆,Q) = 0.


Besides the spectral gap theorem, our main tool is the following.

Theorem (Garland, Ballman–Swiatkowski)If ∆ is a finite, pure d-dimensional, simplicial complex, such that

λ2[lk(σ)] > 1− 1d

for every (d − 2)-dimensional face σ, then Hd−1(∆,Q) = 0.


Using Garland’s method, together with the new spectral gap theorem,we immediately recover the following theorem of Meshulam andWallach.

TheoremLet d ≥ 2 and ε > 0 be fixed and Y ∼ Yd (n,p). Then

P[Hd−1(Y ,Q) = 0]→

1 : p ≥ (d+ε) log n

n


We also get stronger “stopping time” results, analogous to theBollobás–Thomason theorem, and these results are new.


Using Garland’s method, together with the new spectral gap theorem,we immediately recover the following theorem of Meshulam andWallach.

TheoremLet d ≥ 2 and ε > 0 be fixed and Y ∼ Yd (n,p). Then

P[Hd−1(Y ,Q) = 0]→

1 : p ≥ (d+ε) log n

n


We also get stronger “stopping time” results, analogous to theBollobás–Thomason theorem, and these results are new.


The spectral gap theorem and Garland’s method have found severalother applications recently in random topology and geometric grouptheory.

A sharp threshold for π1(Y ) to have Kazhdan’s property (T) (withHoffman and Paquette).Random right angled Coxeter groups are rational duality groups(with Davis).Sharp vanishing thresholds for cohomology of random flagcomplexes.



A sharp threshold for π1(Y ) to have Kazhdan’s property (T) (withHoffman and Paquette).

Random right angled Coxeter groups are rational duality groups(with Davis).Sharp vanishing thresholds for cohomology of random flagcomplexes.



A sharp threshold for π1(Y ) to have Kazhdan’s property (T) (withHoffman and Paquette).Random right angled Coxeter groups are rational duality groups(with Davis).

Sharp vanishing thresholds for cohomology of random flagcomplexes.



A sharp threshold for π1(Y ) to have Kazhdan’s property (T) (withHoffman and Paquette).Random right angled Coxeter groups are rational duality groups(with Davis).Sharp vanishing thresholds for cohomology of random flagcomplexes.


A random flag complex on n = 25 vertices, together with expectedEuler characteristic. Computation and image courtesy of Vidit Nanda.


0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.1 0.2 0.3 0.4 0.5 0.6

rank of homology

edge probability

!0!1!2!3!4!5

A random flag complex on n = 100 vertices, together with expectedEuler characteristic. Computation and image courtesy of AfraZomorodian.


A few words about the geometry of random simplicial complexes.


We know from the Linial–Meshulam theorem that once

p � 2 log nn

,

every cycle is a boundary.

But what is it a boundary of?

Dotterrer, Guth, and I showed that w.h.p. it is not is the boundary ofanything very small.



p � 2 log nn

,






p � 2 log nn

,





Theorem (Dotterrer–Guth–K., in progress)Let

p ≥ nε

n,

where 0 < ε < 1/2 is fixed.Then w.h.p. the 1-cycle [123] in Y ∼ Y (n,p) is not the boundary of anysubcomplex with less than ≈ n2−2ε faces.


Corollary (Dotterrer–Guth–K., in progress)Let 0 < ε < 1/2 be fixed. There exist 2-dimensional simplicialcomplexes with n vertices, and n2+ε two-dimensional faces, and suchthat the smallest 2-cycles contain at least ≈ n2−2ε faces.

Conversely, this is essentially best possible—every 2-dimensionalsimplicial complex with this many fases has a 2-cycle with at most≈ n2−2ε faces.

(By “2-cycle”, we mean a nontrivial class in H2(_,Z/2).)


Future directions


Random geometric complexes

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

β0β1β2β3β4β5

homology

threshold radius r


Toward a general theory of random homology?


Thanks for your time and attention!


Acknowledgements

Collaborators: Eric Babson, Dominic Dotterrer, Larry Guth, ChrisHoffman, Elliot Paquette

Institutional support: Stanford University, Institute for AdvancedStudy, Ohio State University, IMA

Funding: Alfred P. Sloan Foundation, DARPA #N66001-12-1-4226,NSF # CCF-1017182


References

C. Hoffman, M. Kahle, and E. Paquette, Spectral gaps of randomgraphs and applications (submitted)M. Kahle, Topology of random simplicial complexes: a survey (toappear in AMS Contemp. Vol. in Math.)C. Hoffman, M. Kahle, and E. Paquette, The threshold for integerhomology in random d-complexes (submitted)M. Kahle, Sharp vanishing thresholds for cohomology of randomflag complexes (to appear in Annals of Math.)E. Babson, C. Hoffman, and M. Kahle, The fundamental group ofrandom 2-complexes (J. Amer. Math. Soc. 24 (2011), no. 1, p.1–28)


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