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

June 1994 Monday (05/26/2014)

June 1994 Incremental ‘s Monday (05/26/2014)

June 1994 Incremental ‘s Monday (05/26/2014)

June 1994 Incremental ‘s 2002 Topological Persistence Monday (05/26/2014)

June 1994 Incremental ‘s 2002 Topological Persistence 2005 Computing P.H. Monday (05/26/2014)

June 1994 Incremental ‘s 2002 Topological Persistence 2005 Computing P.H Extended Persistence Monday (05/26/2014)

June 1994 Incremental ‘s 2002 Topological Persistence 2005 Computing P.H Extended Persistence 2009 Zig-Zag Persistence … Monday (05/26/2014)

June 1994Monday (05/26/2014) Incremental ‘s 2002 Topological Persistence 2005 Computing P.H Extended Persistence 2009 Zig-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 problem up 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?

* Some times

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

Thanks!!