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Topological Phase Transitions
Robert Adler
Electrical Engineering
Technion ndash Israel Institute of Technology
Stochastic Geometry ndash Poitiers ndash August 2015
Robert Adler
Electrical Engineering
Technion ndash Israel Institute of Technology
and many others
Topological Phase Transitions
(Nature abhors complexity)
ISING MODEL
Starling murmuration Rahat Israel
The wonderful world of geotopology
ALGEBRAIC TOPOLOGY Homology homotopy dimensions of groups Betti numbers persistence
DIFFERENTIAL TOPOLOGY Curvature forms Betti numbers Morse theory integration Lipschitz-Killing curvatures
INTEGRAL GEOMETRY Convexity convex ring kinematic formulae Minkowski functionals
SIMPLICIAL TOPOLOGY Simplices complexes cycles numbers of simplices Betti numbers
9
A 1-slide course on Gaussian random fields
On a topological space M (maybe a stratified Riemannian manifold)
Then take independent Gaussian random variables
and define a Gaussian random field
choose a set of dense functions on M (eg Eigenfunctions of the Laplacian)
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Topological Phase Transitions
Robert Adler
Electrical Engineering
Technion ndash Israel Institute of Technology
Stochastic Geometry ndash Poitiers ndash August 2015
Robert Adler
Electrical Engineering
Technion ndash Israel Institute of Technology
and many others
Topological Phase Transitions
(Nature abhors complexity)
ISING MODEL
Starling murmuration Rahat Israel
The wonderful world of geotopology
ALGEBRAIC TOPOLOGY Homology homotopy dimensions of groups Betti numbers persistence
DIFFERENTIAL TOPOLOGY Curvature forms Betti numbers Morse theory integration Lipschitz-Killing curvatures
INTEGRAL GEOMETRY Convexity convex ring kinematic formulae Minkowski functionals
SIMPLICIAL TOPOLOGY Simplices complexes cycles numbers of simplices Betti numbers
9
A 1-slide course on Gaussian random fields
On a topological space M (maybe a stratified Riemannian manifold)
Then take independent Gaussian random variables
and define a Gaussian random field
choose a set of dense functions on M (eg Eigenfunctions of the Laplacian)
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Robert Adler
Electrical Engineering
Technion ndash Israel Institute of Technology
and many others
Topological Phase Transitions
(Nature abhors complexity)
ISING MODEL
Starling murmuration Rahat Israel
The wonderful world of geotopology
ALGEBRAIC TOPOLOGY Homology homotopy dimensions of groups Betti numbers persistence
DIFFERENTIAL TOPOLOGY Curvature forms Betti numbers Morse theory integration Lipschitz-Killing curvatures
INTEGRAL GEOMETRY Convexity convex ring kinematic formulae Minkowski functionals
SIMPLICIAL TOPOLOGY Simplices complexes cycles numbers of simplices Betti numbers
9
A 1-slide course on Gaussian random fields
On a topological space M (maybe a stratified Riemannian manifold)
Then take independent Gaussian random variables
and define a Gaussian random field
choose a set of dense functions on M (eg Eigenfunctions of the Laplacian)
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
ISING MODEL
Starling murmuration Rahat Israel
The wonderful world of geotopology
ALGEBRAIC TOPOLOGY Homology homotopy dimensions of groups Betti numbers persistence
DIFFERENTIAL TOPOLOGY Curvature forms Betti numbers Morse theory integration Lipschitz-Killing curvatures
INTEGRAL GEOMETRY Convexity convex ring kinematic formulae Minkowski functionals
SIMPLICIAL TOPOLOGY Simplices complexes cycles numbers of simplices Betti numbers
9
A 1-slide course on Gaussian random fields
On a topological space M (maybe a stratified Riemannian manifold)
Then take independent Gaussian random variables
and define a Gaussian random field
choose a set of dense functions on M (eg Eigenfunctions of the Laplacian)
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Starling murmuration Rahat Israel
The wonderful world of geotopology
ALGEBRAIC TOPOLOGY Homology homotopy dimensions of groups Betti numbers persistence
DIFFERENTIAL TOPOLOGY Curvature forms Betti numbers Morse theory integration Lipschitz-Killing curvatures
INTEGRAL GEOMETRY Convexity convex ring kinematic formulae Minkowski functionals
SIMPLICIAL TOPOLOGY Simplices complexes cycles numbers of simplices Betti numbers
9
A 1-slide course on Gaussian random fields
On a topological space M (maybe a stratified Riemannian manifold)
Then take independent Gaussian random variables
and define a Gaussian random field
choose a set of dense functions on M (eg Eigenfunctions of the Laplacian)
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
The wonderful world of geotopology
ALGEBRAIC TOPOLOGY Homology homotopy dimensions of groups Betti numbers persistence
DIFFERENTIAL TOPOLOGY Curvature forms Betti numbers Morse theory integration Lipschitz-Killing curvatures
INTEGRAL GEOMETRY Convexity convex ring kinematic formulae Minkowski functionals
SIMPLICIAL TOPOLOGY Simplices complexes cycles numbers of simplices Betti numbers
9
A 1-slide course on Gaussian random fields
On a topological space M (maybe a stratified Riemannian manifold)
Then take independent Gaussian random variables
and define a Gaussian random field
choose a set of dense functions on M (eg Eigenfunctions of the Laplacian)
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
9
A 1-slide course on Gaussian random fields
On a topological space M (maybe a stratified Riemannian manifold)
Then take independent Gaussian random variables
and define a Gaussian random field
choose a set of dense functions on M (eg Eigenfunctions of the Laplacian)
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Excursion sets
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
11
The Gaussian Kinematic Formula
Theorem
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
12
GKF - Examples
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
13
GKF and phase transitions
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
The separation of homologies
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
15
An example
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Spin glasses and Gaussian critical points
METHOD OF PROOF bull Rice formula for mean bull Exploitation of specific covariance bull Variance for concentration
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
18
The Euler characteristic heuristic
As well as a general theory of the critical points a Gaussian and related via Kac-Slepian models
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
20
The Čech Complex
bullTake a set of vertices P (0-simplexes)
bullDraw balls with radius
bullIntersection of 2 balls an edge (1-simplex)
bullIntersection of 3 balls a triangle (2-simplex)
bullIntersection of n balls a (n-1)-simplex
NERVE THEOREM The Cech complex and union of balls have the same homology
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
E(EC) for Cech complexes on Td
Decreusefond Ferraz Randriam Vergne by counting simplices
2 Poisson process
1 Points (n) chosen at random
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
E(EC) for Cech complexes
n=100d=3
n=100d=8
n=100d=6
n=100d=10
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Cech complex on 1000 points in [01]3
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
24
Phase transitionspersistence for complexes
Dust or subcritical phase
Thermodynamic or critical or percolation phase
Connectivity or supercritical phase
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Theory MK EM OB YD inter alia
fd Dust subcritical
Critical percolative thermodynamic
Dense supercrictical
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Recall = critical points p with
26
Three phases of rigorous results (OB)
bull
bull
bull
Theorem
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
27
Subcritical (dust) phase
Theorem
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Erdos-Renyi graphs and complexes
The graph is on n vertices where each edge appears with probability p independently for each edge
Traditionally only the graph structure is studied The literature is extensive and rich
This is a mean field model Unrealistic but mathematically tractable and often representative
Increasing p increases the topological nature of the graph
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Linial-Meshulam model
bull X(dnp) A generalization of the Erdos-Renyi graph G(np)
bull Start with a full (d minus 1)-dimensional skeleton
bull For each d-dimensional face independently and with probability p decide whether to include it in X(dnp) or not
In the evolution of these complexes with d ge 2 the first occurring cycle is almost surely either
bull 11133231113323 The boundary of a (d + 1)-dimensional simplex or
bull 11133231113323 A cycle that includes Ω(n^d) faces of dimension d
Consequently when d ge 2 different kinds of phase transitions occur
Costa and
Farber and
phase diagrams
Kahle Topology of
random simplicial
complexes a survey
Bobrowski and Kahle
Topology of random
geometric complexes
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
31
Different local structures
Simple perturbed lattice Poisson point process Cox point process
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
32
(Un) Perturbed lattices
Structured point processes lead to a narrower range of topological behaviour
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Models for clouds of interacting points
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Growth of Betti numbers (YD)
Betti numbers taken over the Cech complex with radii rn over growing regions of the form
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Two fluctuation theorems (DY ES)
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
bullDistributions with compact support on
bullDistributions with unbounded support on
Core Crackle
Draw random balls with a fixed radius
What is crackle and why would one care
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Crackle - Definitions
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
How Power-Law Noise Crackles
core
Theorem
where
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
How Exponential Noise Crackles
core
Theorem
where
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
Gaussian Noise Does Not Crackle
Theorem
core no crackle
Recall
So
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
42
Postdoctoral FellowshiPs at the
technion
The preceding slides were brought to you by
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
43
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
44
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
In collaboration with
bull Omer Bobrowski bull Moshe Cohen bull Sunder Ram Krishnan bull Anthea Monod bull Gregory Naitzat bull Takashi Owada bull Tony Reiser bull Yonatan Rosmarin bull Gennady Samorodnitsky bull Eliran Subag bull Jonathan Taylor bull Gugan Thoppe bull Shmuel Weinberger bull Dhandapani Yogeshwaran
![Mixingpropertiesandthechromaticnumber …alexlub/PAPERS/Mixing properties and...shown by Lubotzky and Meshulam in [LuMe] (see [GuLu, Section 4.1] for a ”general principle” of this](https://img.dokumen.tips/doc/110x75/5f359a606f721f6eb834d0f9/mixingpropertiesandthechromaticnumber-alexlubpapersmixing-properties-and-shown.jpg)
![NearlyCompleteGraphsDecomposableintoLargeInduced ......Yet another construction was obtained by Birk, Linial and Meshulam [10], and in an improved formby Meshulam [20]. For the application](https://img.dokumen.tips/doc/110x75/5f3599bba47bd72bf30a3da1/nearlycompletegraphsdecomposableintolargeinduced-yet-another-construction.jpg)
![Nearly Complete Graphs Decomposable into Large …...Yet another construction was obtained by Birk, Linial and Meshulam [11], and in an improved form by Meshulam [22]. For the application](https://img.dokumen.tips/doc/110x75/5f3599bba47bd72bf30a3da3/nearly-complete-graphs-decomposable-into-large-yet-another-construction-was.jpg)
![Research on modulation recognition with ensemble learning...k. 2.4 Renyi entropy According to the reference [22], the Renyi entropy is de-fined by: RαðÞ¼p 1 1−α log 2 P i pα](https://img.dokumen.tips/doc/110x75/608a930a83e9cb1d8b11ab40/research-on-modulation-recognition-with-ensemble-learning-k-24-renyi-entropy.jpg)
