Random topology and geometry

Transcription

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

2 Introduction

3 I predict a new subject of statistical topology. Rather than count the number of holes, Betti numbers, etc., one will be more interested in the distribution of such objects on noncompact manifolds as one goes out to infinity. Isadore Singer, 2004.

4 Randomness models the natural world.

5 Randomness models the natural world. We need probability in order to quantify the significance of topological features detected with topological data analysis.

6 Randomness models the natural world. We need probability in order to quantify the significance of topological features detected with topological data analysis. Certain situations in physics seem to be well modeled by random topology: cosmology, black holes, quantum gravity, etc.

7 Randomness models the natural world. We need probability in order to quantify the significance of topological features detected with topological data analysis. Certain situations in physics seem to be well modeled by random topology: cosmology, black holes, quantum gravity, etc. Can we explain why so many groups / manifolds / simplicial complexes / etc. seem to have a certain topological property?

8 Randomness models the natural world. We need probability in order to quantify the significance of topological features detected with topological data analysis. Certain situations in physics seem to be well modeled by random topology: cosmology, black holes, quantum gravity, etc. Can we explain why so many groups / manifolds / simplicial complexes / etc. seem to have a certain topological property? E.g. many simplicial complexes and posets arising in combinatorics are homotopy equivalent to bouquets of spheres. But why does this happen so often?

9 The probabilistic method provides existence proofs.

10 The probabilistic method provides existence proofs. Ramsey theory and extremal graph theory e.g. Erdős

11 The probabilistic method provides existence proofs. Ramsey theory and extremal graph theory e.g. Erdős Geometric group theory e.g. Gromov, Żuk

12 The probabilistic method provides existence proofs. Ramsey theory and extremal graph theory e.g. Erdős Geometric group theory e.g. Gromov, Żuk Metric geometry e.g. Bourgain

13 The probabilistic method provides existence proofs. Ramsey theory and extremal graph theory e.g. Erdős Geometric group theory e.g. Gromov, Żuk Metric geometry e.g. Bourgain Expander graphs e.g. Pinsker, Barzdin & Kolmogorov

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15 Some earlier work:

16 Some earlier work: Early mention by Milnor 1964

17 Some earlier work: Early mention by Milnor 1964 Gaussian fields on manifolds Adler and Taylor, 2003.

18 Some earlier work: Early mention by Milnor 1964 Gaussian fields on manifolds Adler and Taylor, Random triangulated surfaces Pippenger and Schleich, 2006.

19 Some earlier work: Early mention by Milnor 1964 Gaussian fields on manifolds Adler and Taylor, Random triangulated surfaces Pippenger and Schleich, Random 3-manifolds Dunfield and Thurston, 2006.

20 Some earlier work: Early mention by Milnor 1964 Gaussian fields on manifolds Adler and Taylor, Random triangulated surfaces Pippenger and Schleich, Random 3-manifolds Dunfield and Thurston, Random planar linkages Farber and Kappeller, 2007.

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

22 Random graphs

23 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.

24 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 according to this distribution.

25 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 according to this distribution. It is often useful to think of a stochastic process associated with G(n, p) where edges are added one at a time.

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47 Usually p is a function of n and n.

48 Usually p is a function of n and n. We are often interested in thresholds for graph properties. An archetypal example is the Erdős Rényi theorem.

49 Theorem (Erdős Rényi, 1959) Let ɛ > 0 be fixed and G G(n, p). Then 1 : p (1 + ɛ) log n/n P[G is connected] 0 : p (1 ɛ) log n/n

50 Theorem (Erdős Rényi, 1959) Let ɛ > 0 be fixed and G G(n, p). Then 1 : p (1 + ɛ) log n/n P[G is connected] 0 : p (1 ɛ) log n/n They actually proved a slightly sharper result.

51 Theorem (Erdős Rényi, 1959) Let c R be fixed and G G(n, p). If p = log n + c, n then β 0 (G) is asymptotically Poisson distributed with mean e c and in particular P[G is connected] e e c as n.

52 The first step is to show that if p log n/n, then the probability that there are any components of order i, with 2 i n/2 tends to 0 as n.

53 The first step is to show that if p log n/n, then the probability that there are any components of order i, with 2 i n/2 tends to 0 as n. A union bound argument shows that it is sufficient to show that as n. n/2 k=2 ( ) n k k 2 p k 1 (1 p) k(n k) 0, k

54 The first step is to show that if p log n/n, then the probability that there are any components of order i, with 2 i n/2 tends to 0 as n. A union bound argument shows that it is sufficient to show that as n. n/2 k=2 ( ) n k k 2 p k 1 (1 p) k(n k) 0, k So if p log n/n, then w.h.p. G(n, p) consists of a giant component and isolated vertices.

55 Now set p = (log n + c)/n, where c R is fixed. By linearity of expectation, 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 as n. E[V ] e c,

56 Now set p = (log n + c)/n, where c R is fixed. By linearity of expectation, 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 as n. E[V ] e c, By computing the higher moments, or by Stein s method, one can show that V approaches a Poisson distribution with mean e c.

57 Comments:

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

59 Comments: The proof is phrased in terms of cohomology rather than in terms of homology. The ultimate obstruction to connectivity is isolated vertices. A theorem of Bollobás and Thomason takes this idea to its natural conclusion.

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

61 The normalized graph Laplacian L of a connected graph H is the matrix L = I D 1/2 AD 1/2, where I is the identity matrix, D is the diagonal matrix with degrees down the diagonal, and A is the adjacency matrix.

62 The normalized graph Laplacian L of a connected graph H is the matrix L = I D 1/2 AD 1/2, where I is the identity matrix, D is the diagonal matrix with degrees down the diagonal, and A is the adjacency matrix. The eigenvalues satisfy 0 = λ 1 < λ 2... λ n 2, and λ 2 is often called the spectral gap of the graph.

63 The normalized graph Laplacian L of a connected graph H is the matrix L = I D 1/2 AD 1/2, where I is the identity matrix, D is the diagonal matrix with degrees down the diagonal, and A is the adjacency matrix. The eigenvalues satisfy 0 = λ 1 < λ 2... λ n 2, and λ 2 is often called the spectral gap of the graph. If {H i } is a sequence of graphs such that #V (H i ) and such that lim inf λ 2 [H i ] > 0, as i, we say that {H i } is an expander family.

64 Theorem (Hoffman, K., Paquette, 2013) Fix k 0 and let ω arbitrarily slowly. Let 0 = λ 1 λ 2 λ n 2 be the eigenvalues of the normalized graph Laplacian of G G(n, p). If then p (k + 1) log n + ω, n C 1 np λ C 2 λ n 1 + np, with probability at least 1 o ( n k).

65 Random simplicial complexes

66 Linial and Meshulam defined Y 2 (n, p) to be the probability space of 2-dimensional simplicial complexes with vertex set [n], edge set ( [n]) 2, and such that each 2-face has probability p.

67 More generally, Meshulam and Wallach defined the random d-dimensional simplicial complex Y d (n, p) to be the probability distribution over all simplicial complexes on n vertices with complete d 1-skeleton, and such that every d-dimensional face is included with probability p, independently.

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78 The following theorems generalize the Erdős Rényi theorem to higher dimensions, and inspired a lot of future work by various teams of researchers.

79 The following theorems generalize the Erdős Rényi theorem to higher dimensions, and inspired a lot of future work by various teams of researchers. Theorem (Linial Meshulam, 2006) Let ɛ > 0 be fixed and Y Y 2 (n, p). Then (2+ɛ) log n 1 : p P[H 1 n (Y, Z/2) = 0] 0 : p (2 ɛ) log n n

80 Theorem (Meshulam Wallach, 2008) Let d 2, l 2, and ɛ > 0 be fixed and Y Y d (n, p). Then (d+ɛ) log n 1 : p P[H d 1 n (Y, Z/l) = 0] 0 : p (d ɛ) log n n

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

82 The threshold for simple connectivity is much larger.

83 The threshold for simple connectivity is much larger. Theorem (Babson Hoffman K., 2011) Let ɛ > 0 be fixed and Y Y 2 (n, p). Then 1 : p nɛ n P[π 1 (Y ) = 0] 0 : p n ɛ n

84 We recently revisited homology vanishing theorems for random d-dimensional complexes.

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

86 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 λ 2 [lk(σ)] > 1 1 d for every (d 2)-dimensional face σ, then H d 1 (, Q) = 0.

87 Using Garland s method, together with the new spectral gap theorem, we immediately recover the following theorem of Meshulam and Wallach. Theorem Let d 2 and ɛ > 0 be fixed and Y Y d (n, p). Then 1 : p P[H d 1 (Y, Q) = 0] 0 : p (d+ɛ) log n n (d ɛ) log n n

88 Using Garland s method, together with the new spectral gap theorem, we immediately recover the following theorem of Meshulam and Wallach. Theorem Let d 2 and ɛ > 0 be fixed and Y Y d (n, p). Then 1 : p P[H d 1 (Y, Q) = 0] 0 : p (d+ɛ) log n n (d ɛ) log n n We also get stronger stopping time results, analogous to the Bollobás Thomason theorem, and these results are new.

89 The spectral gap theorem and Garland s method have found several other applications recently in random topology and geometric group theory.

90 The spectral gap theorem and Garland s method have found several other applications recently in random topology and geometric group theory. A sharp threshold for π 1 (Y ) to have Kazhdan s property (T) (with Hoffman and Paquette).

91 The spectral gap theorem and Garland s method have found several other applications recently in random topology and geometric group theory. A sharp threshold for π 1 (Y ) to have Kazhdan s property (T) (with Hoffman and Paquette). Random right angled Coxeter groups are rational duality groups (with Davis).

92 The spectral gap theorem and Garland s method have found several other applications recently in random topology and geometric group theory. A sharp threshold for π 1 (Y ) to have Kazhdan s property (T) (with Hoffman and Paquette). Random right angled Coxeter groups are rational duality groups (with Davis). Sharp vanishing thresholds for cohomology of random flag complexes.

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

94 rank of homology β 0 β 1 β 2 β 3 β 4 β edge probability A random flag complex on n = 100 vertices, together with expected Euler characteristic. Computation and image courtesy of Afra Zomorodian.

95 A few words about the geometry of random simplicial complexes.

96 We know from the Linial Meshulam theorem that once every cycle is a boundary. p 2 log n, n

97 We know from the Linial Meshulam theorem that once every cycle is a boundary. But what is it a boundary of? p 2 log n, n

98 We know from the Linial Meshulam theorem that once every cycle is a boundary. But what is it a boundary of? p 2 log n, n Dotterrer, Guth, and I showed that w.h.p. it is not is the boundary of anything very small.

99 Theorem (Dotterrer Guth K., in progress) Let p nɛ n, where 0 < ɛ < 1/2 is fixed. Then w.h.p. the 1-cycle [123] in Y Y (n, p) is not the boundary of any subcomplex with less than n 2 2ɛ faces.

100 Corollary (Dotterrer Guth K., in progress) Let 0 < ɛ < 1/2 be fixed. There exist 2-dimensional simplicial complexes with n vertices, and n 2+ɛ two-dimensional faces, and such that the smallest 2-cycles contain at least n 2 2ɛ faces. Conversely, this is essentially best possible every 2-dimensional simplicial complex with this many fases has a 2-cycle with at most n 2 2ɛ faces. (By 2-cycle, we mean a nontrivial class in H 2 (_, Z/2).)

101 Future directions

102 Random geometric complexes homology β 0 β 1 β 2 β 3 β 4 β threshold radius r

103 Toward a general theory of random homology?

104 Thanks for your time and attention!

105 Acknowledgements Collaborators: Eric Babson, Dominic Dotterrer, Larry Guth, Chris Hoffman, Elliot Paquette Institutional support: Stanford University, Institute for Advanced Study, Ohio State University, IMA Funding: Alfred P. Sloan Foundation, DARPA #N , NSF # CCF

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