1. Matthew L. WrightMatthew L. Wright Institute for Mathematics and its ApplicationsInstitute for Mathematics and its Applications University of MinnesotaUniversity of Minnesota

2. 4 For a polyhedron, ? – 6+ 4= 2 verticesedgesfaces (# vertices) – (# edges) + (# faces) =

3. For a polyhedron, ? (# vertices) – (# edges) + (# faces) =

4. For a polyhedron, ? (# vertices) – (# edges) + (# faces) = 8– 12+ 6= 2 verticesedgesfaces

5. For a polyhedron, ? (# vertices) – (# edges) + (# faces) =

6. For a polyhedron, ? (# vertices) – (# edges) + (# faces) = The Euler characteristic of a polyhedron is 2.

7. Euler Characteristic, more generally: Finite set Finite graph, no loops Finite graph, with loops Compact planar region = 8= 8 – 5 = 3= 8 – 6 = 2= 8 – 12 + 3 = -1 (number of connected components) – (number of holes)

9. Nota Bene Euler Characteristic is independent of decomposition: = 1 …but is sensitive to open/closed sets: = 0 = 1 = -1

10. Key Property This property is called additivity, or the inclusion-exclusion principle.

12. Integral with respect to Euler Characteristic

13. 1 2 3

14. Preferred in practice! Proof: telescoping sums Now integrate.

15. 2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4

16. 2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4

17. 2 2 2 2 2 3 3 3 1 1 1 1 1 1 0 0 0 0 2 1 1 1 1 2 2 2 3 3 4 4 compute: There are 7 sets! sum 2 1 3 4 5 6 7

18. Proof: Therefore,

19. Application: Euler integration is useful for enumeration problems (Baryshnikov and Ghrist).

20. Can we count by sensors? I count one. sensors network I count none.

21. Can we count by sensors? ??? ? ? ?

22. How can we compute Euler integrals over networks?

23. Computation:

26. Example: Start with a grayscale digital image.

27. 246 635 52 25 164 236 31 56 42155

28. 246 635 52 25 164 236 31 56 42155

29. 246 635 52 25 164 236 31 56 42155 threshold Euler characteristic

30. 246 635 52 25 164 236 31 56 42155 threshold Euler characteristic

31. 246 635 52 25 164 236 31 56 42155 threshold Euler characteristic Euler characteristic graph

32. Richardson and Werman used the Euler characteristic graph to classify 3D objects. a catcurvature function on the cat curvature above threshold 0.05

33. Richardson and Werman used the Euler characteristic graph to classify 3D objects.

34. The Euler characteristic curve helps distinguish between similar artifacts from two sites.

36. Euler characteristic is a simple topological invariant. Homology is a much more sophisticated topological invariant. Betti numbers Connection:

37. Persistent Homology e.g. components, holes, graph structure e.g. set of discrete points, with a metric Persistent homology is an algebraic method for discerning topological features of data.

38. Example: What topological features does the following data exhibit? Problem: Discrete points have trivial topology.

39. Idea: Connect nearby points.

40. Idea: Connect nearby points, build a simplicial complex. 3. Fill in complete simplices.

42. …then we detect noise.

46. A collection of bars is a barcode.

49. ?