Computational Algebraic Topology - Math User Home …hankx003/docs/... · Review Basic Deﬁnitions...

ReviewBasic Definitions

Example

Computational Algebraic Topology

Robert Hank

Department of MathematicsUniversity of Minnesota

Junior Colloquium, 04/09/2012

Robert Hank Computational Algebraic Topology

Example

Outline

1 ReviewPhilosophy of Algebraic TopologySimplicial Homology

2 Basic DefinitionsC̆ech complexPoint CloudsComplexes of Point CloudsPersistence

3 Example

Example

Outline

3 Example

Example

Outline

3 Example

Example

Philosophy of Algebraic TopologySimplicial Homology

Outline

3 Example

Example

We Prefer Algebra over Topology

Use algebra to distinguish spaces

Homotopy groupsHomology groupsCohomology groupsCohomology ring structureOther homology and cohomology theories (SHT, K-theory)

Example

Develop Tools

We want to compute these algebraic structures

Develop theoretical toolsPostnikov towersSpectral sequencesStable homotopy theory“ad-hoc”

Algorithmic/mechanical approach

Example

Develop Tools

Example

Develop Tools

Example

Develop Tools

Example

Develop Tools

Example

Develop Tools

Example

Develop Tools

Example

Outline

3 Example

Example

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.

Example

Cell Complex

Start with 0-cells = points

Example

Cell Complex

Attach 1-cells = edges

Example

Cell Complex

Attach 2-cells = faces

Example

Cell Complex

And continue...

Example

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

Example

In our example

Example

In our example

Example

In our example

Example

In our example

Example

Boundary Maps ∂

We order the verticesGet boundary maps

∂ : Cn(X )→ Cn−1(X )

In general

∂[v0, . . . , vn] =n∑

(−1)i [v0, . . . , v̂i , . . . , vn]

Example

Boundary Maps ∂

∂ : Cn(X )→ Cn−1(X )

In general

∂[v0, . . . , vn] =n∑

(−1)i [v0, . . . , v̂i , . . . , vn]

Example

Boundary Maps ∂

∂ : Cn(X )→ Cn−1(X )

In general

∂[v0, . . . , vn] =n∑

(−1)i [v0, . . . , v̂i , . . . , vn]

Example

Boundary Maps ∂

In our exampleOrder is v0 < v1 < v2 < v3

∂a = v1 − v0

∂F = b − c + a

Example

Boundary Maps ∂

∂a = v1 − v0

∂F = b − c + a

Example

Boundary Maps ∂

∂a = v1 − v0

∂F = b − c + a

Example

Boundary Maps ∂

∂a = v1 − v0

∂F = b − c + a

Example

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 )

Example

Homology Groups

· · · → Cn+1(X )∂−→ Cn(X )

∂−→ Cn−1(X )→ · · ·

Hn(X ,R) =ker(∂ : Cn(X )→ Cn−1(X )

)im(∂ : Cn+1(X )→ Cn(X )

Example

Homology Groups

· · · → Cn+1(X )∂−→ Cn(X )

∂−→ Cn−1(X )→ · · ·

Hn(X ,R) =ker(∂ : Cn(X )→ Cn−1(X )

)im(∂ : Cn+1(X )→ Cn(X )

Example

Computation

QuestionCan we program a machine to do all of this for us?

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.

5 As long as R is a field

Example

Computation

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.

5 As long as R is a field

Example

Computation

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

Example

Computation

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

Example

C̆ech complexPoint CloudsComplexes of Point CloudsPersistence

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 .

Example

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 .

Example

Outline

3 Example

Example

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= ∅

Example

Useful Result

Ua0 ∩ · · · ∩ Uan 6= ∅

Example

Useful Result

Ua0 ∩ · · · ∩ Uan 6= ∅

Example

Useful Result

Ua0 ∩ · · · ∩ Uan 6= ∅

Example

Useful Result

Ua0 ∩ · · · ∩ Uan 6= ∅

Example

Useful Result

TheoremSuppose

U is finiteArbitrary intersections

n⋂i=1

are either contractible or empty.

Then C(U) is homotopy equivalent to X.

Example

Useful Result

TheoremSuppose

n⋂i=1

Example

Useful Result

TheoremSuppose

n⋂i=1

Example

Useful Result

TheoremSuppose

n⋂i=1

are either contractible or empty.Then C(U) is homotopy equivalent to X.

Example

Outline

3 Example

Example

What is a Point Cloud?

Finite collection of points

Example

Sampling

Given a space X , take a “random sampling”

Example

Sampling

Example

Sampling

Example

Sampling

We would like to recover X from the sample

Example

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

Example

Outline

3 Example

Example

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

Example

Metric

Example

Metric

Example

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.

Example

Vietoris-Rips

Example

Vietoris-Rips

Example

Vietoris-Rips

Example

Vietoris-Rips

Example

Vietoris-Rips

Example

Vietoris-Rips

Example

Vietoris-Rips

Example

Vietoris-Rips

Problem: If we start with X as the entire space, this complex isreally large

Example

Vietoris-Rips

Problem: If we start with X as the entire space, this complex isreally large

Example

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

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

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

Example

Witness Complexes

Example

Witness Complexes

Example

Witness Complexes

Example

Outline

3 Example

Example

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

Example

C•(X , ε)→ C•(X , ε′)

Example

C•(X , ε)→ C•(X , ε′)

Example

C•(X , ε)→ C•(X , ε′)

Example

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

Example

Barcodes

Example

Barcodes

Example

Barcodes

Example

Barcodes

Example

Barcodes

Example

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

Example

Problems

C•(X ,n)→ C•(X ,n + 1)

Example

Problems

C•(X ,n)→ C•(X ,n + 1)

Example

Problems

C•(X ,n)→ C•(X ,n + 1)

Example

Problems

C•(X ,n)→ C•(X ,n + 1)

Example

Jennifer GambleZigzag PersistenceSunday 10:10amhere!

Example

Example: Image Statistics

Analyze photographsLee, Pedersen, and Mumford in 2003Thank you to Gunnar Carlsson for the images!

Example

Thank you!

Computational Algebraic Topology - Math User Home …hankx003/docs/... · Review Basic Deﬁnitions...

Documents

Transcript of Computational Algebraic Topology - Math User Home …hankx003/docs/... · Review Basic Deﬁnitions...