Obstructions to Compatible Extensions of Mappings

Duke University

Joint with John Harer

Jose Perea

June 1994

20 years!!

Monday(05/26/2014)

June 1994

Incremental ‘s

Monday(05/26/2014)

June 1994

Incremental ‘s

Monday(05/26/2014)

June 1994

Incremental ‘s

2002Topological Persistence

Monday(05/26/2014)

June 1994

Incremental ‘s

2002Topological Persistence

2005Computing

P.H.

Monday(05/26/2014)

June 1994

Incremental ‘s

2002Topological Persistence

2005Computing

P.H.

2008Extended

Persistence

Monday(05/26/2014)

June 1994

Incremental ‘s

2002Topological Persistence

2005Computing

P.H.

2008Extended

Persistence

2009Zig-Zag

Persistence

…

Monday(05/26/2014)

June 1994 Monday(05/26/2014)

Incremental ‘s

2002Topological Persistence

2005Computing

P.H.

2008Extended

Persistence

2009Zig-Zag

Persistence

What have we learned?Study the whole multi-scale object at once

Is not directionality, but compatible choices

…

…

For Point-cloud data:1. Encode multi-scale information in a filtration-like object

2. Make compatible choices across scales

3. Rank significance of such choices

To leverage the power of

the relative-lifting paradigm

and the language of obstruction theory

The Goal:

To leverage the power of

the relative-lifting paradigm

and the language of obstruction theory

The Goal:

For data analysis!

Why do we care?

Useful concepts/invariants can be interpreted this way:

1. The retraction problem:

2. Extending sections:

3. Characteristic classes.

Back to Point-clouds:

Model fitting

Example (model fitting):

(3-circle model)

(Klein bottle model)

Mumford Data

Model fitting

Only birth-like events

Local to global

Example: Compatible extensions of sections

Local to global

Only death-like events

Local to global

Model fitting

Combine the two:

The compatible-extension problem

How do we set it up?

Definition : The diagram

Extends compatibly, if there exist extensionsof the so that .

For instance :

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

Observation:

Relative lifting problemup to homotopy rel

Compatible extension problem

How do we solve it?

Solving the classic extension problem:

The set-up Assume Want

Solving the classic extension problem:

The set-up Assume Want

Solving the classic extension problem:

The set-up Assume Want

Solving the classic extension problem:

Assume Want

The obstruction cocycle

is a cocycle, and

if and only if extends. Moreover, if for some

then there exists a map

so that on , and

Theorem

is a cocycle, and

if and only if extends. Moreover, if for some

then there exists a map

so that on , and

Theorem

Solving the compatible extension problem:

The set-up

Assume

Let for some .

Then is a cocycle,

which is zero if and only if

Theorem I (Perea, Harer)

Theorem II (Perea, Harer)

Let . If

for , then

and extend compatibly.

The upshot:

Once we fix so that ,

then parametrizes the redefinitions of that

extend. Moreover, if a pair ,

satisfies then the redefinitions of and

via and , extend compatibly.

The upshot:

Once we fix so that ,

then parametrizes the redefinitions of that

extend. Moreover, if a pair ,

satisfies then the redefinitions of and

via and , extend compatibly.

Putting everything together

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…

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Example

Can we actually compute this thing?

Can we actually compute this thing?

Yes!!!

Can we actually compute this thing?

Yes!!!*

* Some times

Coming soon:

• Applications to database consistency

• Topological model fitting

• Bargaining/consensus in social networks

Thanks!!