obstructions to compatible extensions of mappings duke university joint with john harer jose perea
TRANSCRIPT
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
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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
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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!!