Contractibility of a Persistence Map Preimage

1. Contractibility of a persistence map preimage(introduction to) Jacek Cyranka* & Konstantin Mischaikow** ATDD 2018 Montana State University *University California, San Diego, CSE Department **HAPPY BIRTHDAY ATDD’18, Contractibility of a persistence map preimage

2. Motivation slide Nice data compression technique: Instead of storing flow data from an experiment/simulation we want to keep only its persistence diagram(s). Nice , but do we provably recover any characteristic of the dynamics after such (huge) data reduction? ATDD’18, Contractibility of a persistence map preimage

3. The setting of one dimensional simplicial complex Our approach: To study the topology of the preimage of the persistence map to allow for applying existing tools from algebraic topology like the Lefshetz fixed point theorem. ATDD’18, Contractibility of a persistence map preimage

4. The sublevel set filtration death ATDD’18, Contractibility of a persistence map preimage birth

5. loc max loc min loc max loc max loc min loc min Set of all ‘vectors’ whose local extrema are in the prescribed order, and have heights: , ATDD’18, Contractibility of a persistence map preimage

6. Preimage of the persistence map J. Curry, The Fiber of the Persistence Map, arXiv:1706.06059. ATDD’18, Contractibility of a persistence map preimage

7. Obvious fact is that is (highly) nonconvex ATDD’18, Contractibility of a persistence map preimage

8. A study of the preimage set topology from now on the vector of local extrema heights is fixed) Let we say that a vector is monotone if the local extrema with heights defined by , appear in positions defined by .

9. Study of the preimage set topology – Step II Example for and What about contractibility of ? ATDD’18, Contractibility of a persistence map preimage 9

10. A case study of applying the Nerve theorem P. Alexandroff. Uber den allgemeinen dimensionsbegriff und seine beziehungen zur elementaren geometrischen anschauung. Mathematische Annalen, 98(1):617–635, 1928. K. Borsuk. On the imbedding of systems of compacta in simplicial complexes. Fund. Math., 35:217–234, 1948. ATDD’18, Contractibility of a persistence map preimage

11. Study of the preimage set topology–Step III The covering of using ’s is too fine, cannot compute all intersections for the Nerve lemma ATDD’18, Contractibility of a persistence map preimage

12. Study of the preimage set topology–Step III Question: How to study intersections of families of ’s ? Solution: Generalize the definition of ’s, and prove is a semi-lattice morphism. ATDD’18, Contractibility of a persistence map preimage

13. Generalization - set of unordered multi-indices ATDD’18, Contractibility of a persistence map preimage

14. Generalization - set of not ordered multi-indices Redefine (generalized) ’s ATDD’18, Contractibility of a persistence map preimage

15. Full characterization of intersections of ’s Our first important result And we characterize all ’s as either empty sets or star-shaped sets ATDD’18, Contractibility of a persistence map preimage

16. Lets apply the theorems from the previous slides ATDD’18, Contractibility of a persistence map preimage

17. Building the nerve complex 2. Compute pairwise intersections and put an edge when contractible 3. Compute intersection of triples and put an edge when contractible 1. For put a node. ATDD’18, Contractibility of a persistence map preimage

18. The story is not yet over, but there is a happy end… HA ! I see that this nerve complex is contractible Great ! But wait, how to generalize the construction of the nerve complex to prove its contractibility in general ? Solution: We can group ’s and define an even more coarse covering, such that the resulting simplicial complex is the full simplex. ATDD’18, Contractibility of a persistence map preimage