Obstructions to Compatible Extensions of Mappings Duke University Joint with John Harer Jose Perea

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

extensions

of 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

…

…

…

…

…

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!!