Obstructions to Compatible Extensions of Mappings

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

2. 20 years!! Monday (05/26/2014) June 1994

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

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

5. 2002 Monday (05/26/2014) Topological Persistence June 1994 Incremental ‘s

6. 2005 2002 Monday (05/26/2014) Topological Persistence Computing P.H. June 1994 Incremental ‘s

7. 2005 2008 2002 Monday (05/26/2014) Topological Persistence Extended Persistence Computing P.H. June 1994 Incremental ‘s

8. … 2005 2008 2009 2002 Monday (05/26/2014) Topological Persistence Extended Persistence Zig-Zag Persistence Computing P.H. June 1994 Incremental ‘s

9. 2005 2008 2009 2002 Topological Persistence Extended Persistence Zig-Zag Persistence Computing P.H. June 1994 Monday (05/26/2014) Incremental ‘s

11. What have we learned? Study the whole multi-scale object at once Is not directionality, but compatible choices … …

12. For Point-cloud data: • Encode multi-scale information in a filtration-like object • Make compatible choices across scales • Rank significance of such choices

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

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

15. Why do we care?

16. Useful concepts/invariants can be interpreted this way: • The retraction problem: • Extending sections: • Characteristic classes.

17. Back to Point-clouds: Model fitting

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

19. Model fitting Only birth-like events

20. Example: Compatible extensions of sections Local to global

21. Only death-like events Local to global

22. Model fitting Local to global

23. Combine the two: The compatible-extension problem

24. How do we set it up?

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

26. For instance :

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

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

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

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

31. Observation: Compatible extension problem Relative lifting problem up to homotopyrel

32. How do we solve it?

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

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

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

36. Solving the classic extension problem: The obstruction cocycle Want Assume

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

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

39. Solving the compatible extension problem: The set-up Assume

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

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

44. 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.

45. 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.

46. Putting everything together

48. … … … … …

49. Example