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

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June 1994 Monday (05/26/2014)

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

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

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

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

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June 1994 Incremental ‘s 2002 Topological Persistence 2005 Computing P.H. 2008 Extended Persistence Monday (05/26/2014)

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June 1994 Incremental ‘s 2002 Topological Persistence 2005 Computing P.H. 2008 Extended Persistence 2009 Zig-Zag Persistence … Monday (05/26/2014)

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June 1994Monday (05/26/2014) Incremental ‘s 2002 Topological Persistence 2005 Computing P.H. 2008 Extended Persistence 2009 Zig-Zag Persistence

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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

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To leverage the power of the relative-lifting paradigm and the language of obstruction theory The Goal:

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To leverage the power of the relative-lifting paradigm and the language of obstruction theory The Goal: For data analysis!

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Why do we care?

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Useful concepts/invariants can be interpreted this way: 1.The retraction problem: 2.Extending sections: 3.Characteristic classes.

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Back to Point-clouds: Model fitting

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Example (model fitting): (3-circle model) (Klein bottle model) Mumford Data

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Model fitting Only birth-like events

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Local to global Example: Compatible extensions of sections

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Local to global Only death-like events

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Local to global Model fitting

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Combine the two: The compatible-extension problem

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How do we set it up?

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Definition : The diagram Extends compatibly, if there exist extensions of the so that.

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For instance :

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Let be the tangent bundle over, and fix classifying maps If then, where Thus, Extend separately but not compatibly

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Let be the tangent bundle over, and fix classifying maps If then, where Thus, Extend separately but not compatibly

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Let be the tangent bundle over, and fix classifying maps If then, where Thus, Extend separately but not compatibly

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Let be the tangent bundle over, and fix classifying maps If then, where Thus, Extend separately but not compatibly

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Observation: Relative lifting problem up to homotopy rel Compatible extension problem

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How do we solve it?

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Solving the classic extension problem: The set-up Assume Want

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Solving the classic extension problem: The set-up Assume Want

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Solving the classic extension problem: The set-up Assume Want

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Solving the classic extension problem: Assume Want The obstruction cocycle

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is a cocycle, and if and only if extends. Moreover, if for some then there exists a map so that on, and Theorem

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is a cocycle, and if and only if extends. Moreover, if for some then there exists a map so that on, and Theorem

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Solving the compatible extension problem: The set-up Assume

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Let for some. Then is a cocycle, which is zero if and only if Theorem I (Perea, Harer)

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Theorem II (Perea, Harer) Let. If for, then and extend compatibly.

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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.

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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.

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Putting everything together

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… … … … …

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Example

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Can we actually compute this thing?

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* Some times

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

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Thanks!!

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