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Obstructions to Compatible Extensions of Mappings. Jose Perea. Duke University. Joint with John Harer. 20 years!!. Monday (05/26/2014). June 1994. Monday (05/26/2014). June 1994. Incremental ‘s . Monday (05/26/2014). June 1994. Incremental ‘s . 2002.

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obstructions to compatible extensions of mappings

Obstructions to Compatible Extensions of Mappings

Jose Perea

Duke University

Joint with John Harer

slide2

20 years!!

Monday

(05/26/2014)

June 1994

slide3

Monday

(05/26/2014)

June 1994

Incremental ‘s

slide4

Monday

(05/26/2014)

June 1994

Incremental ‘s

slide5

2002

Monday

(05/26/2014)

Topological Persistence

June 1994

Incremental ‘s

slide6

2005

2002

Monday

(05/26/2014)

Topological Persistence

Computing

P.H.

June 1994

Incremental ‘s

slide7

2005

2008

2002

Monday

(05/26/2014)

Topological Persistence

Extended

Persistence

Computing

P.H.

June 1994

Incremental ‘s

slide8

2005

2008

2009

2002

Monday

(05/26/2014)

Topological Persistence

Extended

Persistence

Zig-Zag

Persistence

Computing

P.H.

June 1994

Incremental ‘s

slide9

2005

2008

2009

2002

Topological Persistence

Extended

Persistence

Zig-Zag

Persistence

Computing

P.H.

June 1994

Monday

(05/26/2014)

Incremental ‘s

what have we learned
What have we learned?

Study the whole

multi-scale object at once

Is not directionality, but compatible choices

for point cloud data
For Point-cloud data:
  • Encode multi-scale information in a filtration-like object
  • Make compatible choices across scales
  • Rank significance of such choices
the goal
The Goal:

To leverage the power of

the relative-lifting paradigm

and the language of obstruction theory

the goal1
The Goal:

To leverage the power of

the relative-lifting paradigm

and the language of obstruction theory

For data analysis!

useful concepts invariants can be interpreted this way
Useful concepts/invariants can be interpreted this way:
  • The retraction problem:
  • Extending sections:
  • Characteristic classes.
example model fitting
Example (model fitting):

(Klein bottle model)

(3-circle model)

Mumford Data

slide19

Model fitting

Only birth-like events

slide21

Only death-like events

Local to global

slide22

Model fitting

Local to global

combine the two
Combine the two:

The compatible-extension problem

definition the diagram
Definition : The diagram

Extends compatibly, if there exist extensions

of the so that .

slide27

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

slide28

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

slide29

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

slide30

Let be the tangent bundle over , and fix classifying maps

If then , where

Thus,

Extend separately but

not compatibly

slide31

Observation:

Compatible extension problem

Relative lifting problem

up to homotopyrel

solving the classic extension problem3
Solving the classic extension problem:

The obstruction cocycle

Want

Assume

theorem
Theorem

is a cocycle, and

if and only if extends. Moreover, if for some

then there exists a map

so that on , and

theorem1
Theorem

is a cocycle, and

if and only if extends. Moreover, if for some

then there exists a map

so that on , and

theorem i perea harer
Theorem I (Perea, Harer)

Let for some .

Then is a cocycle,

which is zero if and only if

theorem ii perea harer
Theorem II (Perea, Harer)

Let . If

for , then

and extend compatibly.

the upshot
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 upshot1
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.

slide48

coming soon
Coming soon:
  • Applications to database consistency
  • Topological model fitting
  • Bargaining/consensus in social networks