ReviewBasic Definitions
Example
Computational Algebraic Topology
Robert Hank
Department of MathematicsUniversity of Minnesota
Junior Colloquium, 04/09/2012
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces
Homotopy groupsHomology groupsCohomology groupsCohomology ring structureOther homology and cohomology theories (SHT, K-theory)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces
Homotopy groupsHomology groupsCohomology groupsCohomology ring structureOther homology and cohomology theories (SHT, K-theory)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces
Homotopy groupsHomology groupsCohomology groupsCohomology ring structureOther homology and cohomology theories (SHT, K-theory)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces
Homotopy groupsHomology groupsCohomology groupsCohomology ring structureOther homology and cohomology theories (SHT, K-theory)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces
Homotopy groupsHomology groupsCohomology groupsCohomology ring structureOther homology and cohomology theories (SHT, K-theory)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces
Homotopy groupsHomology groupsCohomology groupsCohomology ring structureOther homology and cohomology theories (SHT, K-theory)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Develop Tools
We want to compute these algebraic structures
Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”
Algorithmic/mechanical approach
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Develop Tools
We want to compute these algebraic structures
Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”
Algorithmic/mechanical approach
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Develop Tools
We want to compute these algebraic structures
Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”
Algorithmic/mechanical approach
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Develop Tools
We want to compute these algebraic structures
Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”
Algorithmic/mechanical approach
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Develop Tools
We want to compute these algebraic structures
Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”
Algorithmic/mechanical approach
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Develop Tools
We want to compute these algebraic structures
Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”
Algorithmic/mechanical approach
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Develop Tools
We want to compute these algebraic structures
Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”
Algorithmic/mechanical approach
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Recall the Usual Procedure
1 Create a cell complex X .2 Create a graded R-module Cn(X ; R).3 Turn into a chain complex using the boundary maps ∂.4 Take homology of the complex.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Recall the Usual Procedure
1 Create a cell complex X .2 Create a graded R-module Cn(X ; R).3 Turn into a chain complex using the boundary maps ∂.4 Take homology of the complex.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Recall the Usual Procedure
1 Create a cell complex X .2 Create a graded R-module Cn(X ; R).3 Turn into a chain complex using the boundary maps ∂.4 Take homology of the complex.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Recall the Usual Procedure
1 Create a cell complex X .2 Create a graded R-module Cn(X ; R).3 Turn into a chain complex using the boundary maps ∂.4 Take homology of the complex.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Recall the Usual Procedure
1 Create a cell complex X .2 Create a graded R-module Cn(X ; R).3 Turn into a chain complex using the boundary maps ∂.4 Take homology of the complex.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Cell Complex
v0
1
2
3
v
v
v
Start with 0-cells = points
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Cell Complex
v0
1
2
3
v
v
v
a
b
c d
e
Attach 1-cells = edges
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Cell Complex
Attach 2-cells = faces
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Cell Complex
And continue...
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Translate Into Algebra
Next, we take all our cells and group them together bydimension
Cn(X ,R) = R[n-dimensional cells].
In our example
C0(X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3C1(X ) = Ra⊕ Rb ⊕ Rc ⊕ Rd ⊕ ReC2(X ) = RF ⊕ RG
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Translate Into Algebra
Next, we take all our cells and group them together bydimension
Cn(X ,R) = R[n-dimensional cells].
In our example
C0(X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3C1(X ) = Ra⊕ Rb ⊕ Rc ⊕ Rd ⊕ ReC2(X ) = RF ⊕ RG
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Translate Into Algebra
Next, we take all our cells and group them together bydimension
Cn(X ,R) = R[n-dimensional cells].
In our example
C0(X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3C1(X ) = Ra⊕ Rb ⊕ Rc ⊕ Rd ⊕ ReC2(X ) = RF ⊕ RG
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Translate Into Algebra
Next, we take all our cells and group them together bydimension
Cn(X ,R) = R[n-dimensional cells].
In our example
C0(X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3C1(X ) = Ra⊕ Rb ⊕ Rc ⊕ Rd ⊕ ReC2(X ) = RF ⊕ RG
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Translate Into Algebra
Next, we take all our cells and group them together bydimension
Cn(X ,R) = R[n-dimensional cells].
In our example
C0(X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3C1(X ) = Ra⊕ Rb ⊕ Rc ⊕ Rd ⊕ ReC2(X ) = RF ⊕ RG
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Boundary Maps ∂
We order the verticesGet boundary maps
∂ : Cn(X )→ Cn−1(X )
In general
∂[v0, . . . , vn] =n∑
i=0
(−1)i [v0, . . . , v̂i , . . . , vn]
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Boundary Maps ∂
We order the verticesGet boundary maps
∂ : Cn(X )→ Cn−1(X )
In general
∂[v0, . . . , vn] =n∑
i=0
(−1)i [v0, . . . , v̂i , . . . , vn]
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Boundary Maps ∂
We order the verticesGet boundary maps
∂ : Cn(X )→ Cn−1(X )
In general
∂[v0, . . . , vn] =n∑
i=0
(−1)i [v0, . . . , v̂i , . . . , vn]
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Boundary Maps ∂
In our exampleOrder is v0 < v1 < v2 < v3
∂a = v1 − v0
∂F = b − c + a
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Boundary Maps ∂
In our exampleOrder is v0 < v1 < v2 < v3
∂a = v1 − v0
∂F = b − c + a
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Boundary Maps ∂
In our exampleOrder is v0 < v1 < v2 < v3
∂a = v1 − v0
∂F = b − c + a
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Boundary Maps ∂
In our exampleOrder is v0 < v1 < v2 < v3
∂a = v1 − v0
∂F = b − c + a
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Homology Groups
Prove ∂2 = 0Gives a chain complex
· · · → Cn+1(X )∂−→ Cn(X )
∂−→ Cn−1(X )→ · · ·
Homology of the complex is simplicial homology
Hn(X ,R) =ker(∂ : Cn(X )→ Cn−1(X )
)im(∂ : Cn+1(X )→ Cn(X )
)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Homology Groups
Prove ∂2 = 0Gives a chain complex
· · · → Cn+1(X )∂−→ Cn(X )
∂−→ Cn−1(X )→ · · ·
Homology of the complex is simplicial homology
Hn(X ,R) =ker(∂ : Cn(X )→ Cn−1(X )
)im(∂ : Cn+1(X )→ Cn(X )
)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Homology Groups
Prove ∂2 = 0Gives a chain complex
· · · → Cn+1(X )∂−→ Cn(X )
∂−→ Cn−1(X )→ · · ·
Homology of the complex is simplicial homology
Hn(X ,R) =ker(∂ : Cn(X )→ Cn−1(X )
)im(∂ : Cn+1(X )→ Cn(X )
)
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Computation
QuestionCan we program a machine to do all of this for us?
1 Create a cell complex X .
Yes!
2 Create a graded R-module Cn(X ; R).
Yes
3 Turn into a chain complex using the boundary maps ∂.
Yes
4 Take homology of the complex.
Yes
5 As long as R is a field
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Computation
QuestionCan we program a machine to do all of this for us?
1 Create a cell complex X .
Yes!
2 Create a graded R-module Cn(X ; R).
Yes
3 Turn into a chain complex using the boundary maps ∂.
Yes
4 Take homology of the complex.
Yes
5 As long as R is a field
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Computation
QuestionCan we program a machine to do all of this for us?
1 Create a cell complex X .
Yes!
2 Create a graded R-module Cn(X ; R). Yes3 Turn into a chain complex using the boundary maps ∂. Yes4 Take homology of the complex. Yes5 As long as R is a field
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Philosophy of Algebraic TopologySimplicial Homology
Computation
QuestionCan we program a machine to do all of this for us?
1 Create a cell complex X . Yes!2 Create a graded R-module Cn(X ; R). Yes3 Turn into a chain complex using the boundary maps ∂. Yes4 Take homology of the complex. Yes5 As long as R is a field
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Given a space X , how can we turn over the computation ofhomology to a machine?
Usually have some idea of distance on X , and we use it tocover X .
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Given a space X , how can we turn over the computation ofhomology to a machine?
Usually have some idea of distance on X , and we use it tocover X .
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
NotationU = {Ua}a∈A is an open cover of X
C(U) is the C̆ech complex of UVertex set Aa0, . . . ,an span an n-simplex if
Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
NotationU = {Ua}a∈A is an open cover of X
C(U) is the C̆ech complex of UVertex set Aa0, . . . ,an span an n-simplex if
Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
NotationU = {Ua}a∈A is an open cover of X
C(U) is the C̆ech complex of UVertex set Aa0, . . . ,an span an n-simplex if
Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
NotationU = {Ua}a∈A is an open cover of X
C(U) is the C̆ech complex of UVertex set Aa0, . . . ,an span an n-simplex if
Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
NotationU = {Ua}a∈A is an open cover of X
C(U) is the C̆ech complex of UVertex set Aa0, . . . ,an span an n-simplex if
Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
TheoremSuppose
U is finiteArbitrary intersections
n⋂i=1
Uai
are either contractible or empty.
Then C(U) is homotopy equivalent to X.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
TheoremSuppose
U is finiteArbitrary intersections
n⋂i=1
Uai
are either contractible or empty.
Then C(U) is homotopy equivalent to X.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
TheoremSuppose
U is finiteArbitrary intersections
n⋂i=1
Uai
are either contractible or empty.
Then C(U) is homotopy equivalent to X.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Useful Result
TheoremSuppose
U is finiteArbitrary intersections
n⋂i=1
Uai
are either contractible or empty.Then C(U) is homotopy equivalent to X.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
What is a Point Cloud?
Finite collection of points
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
What is a Point Cloud?
Finite collection of points
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
What is a Point Cloud?
Finite collection of points
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
What is a Point Cloud?
Finite collection of points
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Sampling
Given a space X , take a “random sampling”
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Sampling
Given a space X , take a “random sampling”
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Sampling
Given a space X , take a “random sampling”
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Sampling
We would like to recover X from the sample
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Data
Data samples can also create point cloudsWe would like to understand the shape the data takesInsights into the shape can be very useful for interpretingthe data
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Data
Data samples can also create point cloudsWe would like to understand the shape the data takesInsights into the shape can be very useful for interpretingthe data
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Data
Data samples can also create point cloudsWe would like to understand the shape the data takesInsights into the shape can be very useful for interpretingthe data
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Metric
In practice, our space X will come equipped with a metric dWe can use the metric to construct a complexThis can be programmed so a computer can run thealgorithm
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Metric
In practice, our space X will come equipped with a metric dWe can use the metric to construct a complexThis can be programmed so a computer can run thealgorithm
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Metric
In practice, our space X will come equipped with a metric dWe can use the metric to construct a complexThis can be programmed so a computer can run thealgorithm
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Given a point cloud X , VR(X , ε) is the Vietoris-Rips complexassociated to the parameter ε.
Vertex set is Xx0, . . . , xn span an n-simplex if the distance between anypair is ≤ ε.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Given a point cloud X , VR(X , ε) is the Vietoris-Rips complexassociated to the parameter ε.
Vertex set is Xx0, . . . , xn span an n-simplex if the distance between anypair is ≤ ε.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Given a point cloud X , VR(X , ε) is the Vietoris-Rips complexassociated to the parameter ε.
Vertex set is Xx0, . . . , xn span an n-simplex if the distance between anypair is ≤ ε.
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Problem: If we start with X as the entire space, this complex isreally large
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Vietoris-Rips
Problem: If we start with X as the entire space, this complex isreally large
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
l0, . . . , ln span an n-simplex if d(x , li) ≤ mx + ε for every iThis can be automated and gives smaller complexesEfficiency adjustments give other types of witnesscomplexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
l0, . . . , ln span an n-simplex if d(x , li) ≤ mx + ε for every iThis can be automated and gives smaller complexesEfficiency adjustments give other types of witnesscomplexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
l0, . . . , ln span an n-simplex if d(x , li) ≤ mx + ε for every iThis can be automated and gives smaller complexesEfficiency adjustments give other types of witnesscomplexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Witness Complexes
Solution: Sample X with a set L of “landmark points”Given x ∈ X , let mx be the distance from x to LChoose εx is a “witness” to li if d(x , li) ≤ mx + ε
l0, . . . , ln span an n-simplex if d(x , li) ≤ mx + ε for every iThis can be automated and gives smaller complexesEfficiency adjustments give other types of witnesscomplexes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Outline
1 ReviewPhilosophy of Algebraic TopologySimplicial Homology
2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence
3 Example
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
In all our complexes (C̆ech, Vietoris-Rips, witness), wehave a changing parameter εWe get different complexes C•(X , ε) depending on εBut if ε < ε′ we get an inclusion
C•(X , ε)→ C•(X , ε′)
Eventually, ε is so large that C•(X , ε) doesn’t change
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
In all our complexes (C̆ech, Vietoris-Rips, witness), wehave a changing parameter εWe get different complexes C•(X , ε) depending on εBut if ε < ε′ we get an inclusion
C•(X , ε)→ C•(X , ε′)
Eventually, ε is so large that C•(X , ε) doesn’t change
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
In all our complexes (C̆ech, Vietoris-Rips, witness), wehave a changing parameter εWe get different complexes C•(X , ε) depending on εBut if ε < ε′ we get an inclusion
C•(X , ε)→ C•(X , ε′)
Eventually, ε is so large that C•(X , ε) doesn’t change
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Idea
In all our complexes (C̆ech, Vietoris-Rips, witness), wehave a changing parameter εWe get different complexes C•(X , ε) depending on εBut if ε < ε′ we get an inclusion
C•(X , ε)→ C•(X , ε′)
Eventually, ε is so large that C•(X , ε) doesn’t change
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Barcodes
If our base ring is a field FOur complexes are vector spacesWe have algebraic algorithms to decompose vector spacesMake a countable, order-preserving choice of paramatersε, f : N→ RRepresent the change in vector spaces via barcodes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Barcodes
If our base ring is a field FOur complexes are vector spacesWe have algebraic algorithms to decompose vector spacesMake a countable, order-preserving choice of paramatersε, f : N→ RRepresent the change in vector spaces via barcodes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Barcodes
If our base ring is a field FOur complexes are vector spacesWe have algebraic algorithms to decompose vector spacesMake a countable, order-preserving choice of paramatersε, f : N→ RRepresent the change in vector spaces via barcodes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Barcodes
If our base ring is a field FOur complexes are vector spacesWe have algebraic algorithms to decompose vector spacesMake a countable, order-preserving choice of paramatersε, f : N→ RRepresent the change in vector spaces via barcodes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Barcodes
If our base ring is a field FOur complexes are vector spacesWe have algebraic algorithms to decompose vector spacesMake a countable, order-preserving choice of paramatersε, f : N→ RRepresent the change in vector spaces via barcodes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Barcodes
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Problems
How do we handle multiple parameters changing (e.g. L)?Multidimensional PersistencePersistence relies on having nested maps
C•(X ,n)→ C•(X ,n + 1)
What if we can’t get these? Zigzag Persistence
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Problems
How do we handle multiple parameters changing (e.g. L)?Multidimensional PersistencePersistence relies on having nested maps
C•(X ,n)→ C•(X ,n + 1)
What if we can’t get these? Zigzag Persistence
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Problems
How do we handle multiple parameters changing (e.g. L)?Multidimensional PersistencePersistence relies on having nested maps
C•(X ,n)→ C•(X ,n + 1)
What if we can’t get these? Zigzag Persistence
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Problems
How do we handle multiple parameters changing (e.g. L)?Multidimensional PersistencePersistence relies on having nested maps
C•(X ,n)→ C•(X ,n + 1)
What if we can’t get these? Zigzag Persistence
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
Problems
How do we handle multiple parameters changing (e.g. L)?Multidimensional PersistencePersistence relies on having nested maps
C•(X ,n)→ C•(X ,n + 1)
What if we can’t get these? Zigzag Persistence
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
C̆ech complexPoint CloudsComplexes of Point CloudsPersistence
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Robert Hank Computational Algebraic Topology
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Analyze photographsLee, Pedersen, and Mumford in 2003Thank you to Gunnar Carlsson for the images!
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
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Example: Image Statistics
Analyze photographsLee, Pedersen, and Mumford in 2003Thank you to Gunnar Carlsson for the images!
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Analyze photographsLee, Pedersen, and Mumford in 2003Thank you to Gunnar Carlsson for the images!
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology
ReviewBasic Definitions
Example
Example: Image Statistics
Robert Hank Computational Algebraic Topology