Randomness in Topology: Persistence Diagrams, Euler Characteristics, and Möbius Inversion

Exploring the concept of randomness in topology, this work delves into the fascinating realms of persistence diagrams, Euler characteristics, and Möbius inversion. Jointly presented with Amit Patel, the study uncovers the vast generalization of Möbius inversion as a principle of inclusion-exclusion, touching upon historical figures like August Ferdinand Möbius and innovative combinatorial theories. The visual representations provide a deeper understanding of posets, Möbius functions, examples of inclusion-exclusion, and Möbius inversion applications in persistence diagrams.

• Topology
• Randomness
• Persistence Diagrams
• Möbius Inversion
• Euler Characteristics

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1. PERSISTENCE DIAGRAMS, EULER CHARACTERISTICS, & M BIUS INVERSION Primoz Skraba Randomness in Topology & Its Applications March 21, 2023

2. Outline Joint work with Amit Patel

3. Outline M bius inversion Euler Persistence diagrams characteristic

4. Mbius Inversion M bius inversion is a vast generalization of the principle of inclusion-exclusion.

5. Mbius Inversion M bius inversion is a vast generalization of the principle of inclusion-exclusion. August Ferdinand M bius 1832 Euler s Totient Function 1763 Notation from Gauss 1801 Gian-Carlo Rota 1964 On the foundations of combinatorial theory I

6. Posets Notation:

7. Posets Notation:

8. Mbius Function Example:

9. Example: Inclusion-Exclusion

10. Mbius Inversion Example:

11. MBIUS INVERSION PERSISTENCEDIAGRAMS

12. Persistence Diagrams Classical Persistence:

13. Posets of Intervals Elements: set of all intervals, Ordering: two possibilities (Inclusion ordering) (Product ordering)

14. PD as Mbius Inversion 2007: Cohen-Steiner, Edelsbrunner, Harer 2016: Patel Inclusion ordering

15. PD as Mbius Inversion 2007: Cohen-Steiner, Edelsbrunner, Harer 2016: Patel Inclusion ordering

16. PD as Mbius Inversion 2007: Cohen-Steiner, Edelsbrunner, Harer 2016: Patel Inclusion ordering

17. PD as Mbius Inversion 2007: Cohen-Steiner, Edelsbrunner, Harer 2016: Patel Inclusion ordering

18. PD as Mbius Inversion 2007: Cohen-Steiner, Edelsbrunner, Harer 2016: Patel Inclusion ordering

19. PD as Mbius Inversion 2007: Cohen-Steiner, Edelsbrunner, Harer 2016: Patel Inclusion ordering

20. Multiparameter Persistence Works over any poset Generalized rank invariant, fibered barcode, etc.

21. Birth-Death Function 2016: Henselman-Petrusek, Ghrist 2022: McCleary, Patel Product ordering

22. Birth-Death Function 2016: Henselman-Petrusek, Ghrist 2022: McCleary, Patel Product ordering

23. Birth-Death Function 2016: Henselman-Petrusek, Ghrist 2022: McCleary, Patel Product ordering

24. Kernel Function Product ordering 2022: G len, McCleary

25. Kernel Function Product ordering 2022: G len, McCleary

26. Equivalence 2022: G len, McCleary Birth-Death Function Same Persistence Diagram (up to the diagonal) Kernel Function

27. Applications Bottleneck Stability rank function Duy, Hiraoka, Shirai: Limit Theorems for Persistence Diagrams birth-death function Appearance: Proof of convergence

28. Applications Bottleneck Stability rank function Duy, Hiraoka, Shirai: Limit Theorems for Persistence Diagrams birth-death function Appearance: Proof of convergence Exploit monotonicity of birth-death function

29. MBIUS FUNCTION EULER CHARACTERISTIC

30. Order Complex

31. Euler Characteristic Theorem [Hall 1928 Rota 1964]

32. Related Work Reference Stanley, Enumerative combinatorics, Vol. 1 Wachs, Poset Topology: Tools and Applications Highlights Rota, Folkman, ivaljevi , Ziegler, Kozlov, Walker, Stanley, Crapo, Bj rner,

33. PERSISTENCE DIAGRAMS EULER CHARACTERISTICS

34. Modules over Posets Recall: Module

35. Modules over Posets Recall: What is the M bius inversion of a vector space? Patel 2016

36. Back to Topology (Simplicial) Cosheaf

37. Rota Cosheaf Example: Rule:

38. Cosheaf Euler Characteristic

39. Mbius Inversion as EC Theorem

40. Mbius Inversion as EC Theorem

41. Key Idea Hall s Theorem [1928]

42. Cosheaf Homology Rather than the Euler characteristic, we can take the homology groups

43. Cosheaf Homology Rather than the Euler characteristic, we can take the homology groups

44. Example

45. 2D Example Does considering homology help? *Example due to Vidit Nanda

46. 2D Example

47. 2D Example

48. GALOIS CONNECTIONS

49. Galois Connections

50. Rotas Galois Connection Theorem Theorem [Rota, G len-McCleary]

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