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!!
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