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ima.umn /2008-2009/ND6.15-26.09

MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 6, 2013: Calculating homology using matrices Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics

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ima.umn /2008-2009/ND6.15-26.09

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  1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 6, 2013: Calculating homology using matrices Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics Applied Mathematical and Computational Sciences, University of Iowa http://www.math.uiowa.edu/~idarcy/AppliedTopology.html

  2. http://www.ima.umn.edu/2008-2009/ND6.15-26.09 http://www.ima.umn.edu/2008-2009/ND6.15-26.09/abstracts.html#8322

  3. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 H0 = Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> H0 = Z0/B0 = <[v1], [v4]> where [v1] = {v1, v2, v3} and [v4] = {v4, v5, v6} e5

  4. Counting number of connected components using homology Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> Use matrices: See Computing Persistent Homology by AfraZomorodian, Gunnar Carlsson

  5. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 : C1  C0 Extend linearly: v4 v6 v1 v3 (e3) = v4 + v5 (e4) = v5 + v6 (e1) = v1 + v2 (e5) = v4 + v6 e5 (e2) = v2 + v3 (Sniei) = niS (ei) o o o o o o o o

  6. Counting number of connected components using homology Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> Use matrices: See Computing Persistent Homology by AfraZomorodian, Gunnar Carlsson

  7. Let e1 = {v1, v2} = Let e2 = {v2, v3} = Let e3 = {v4, v5} = Let e4 = {v5, v6} = Let e5 = {v4, v6} =

  8. v1 = , v2 = , v3 = , v4 = , v5 = v6 =

  9. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 (e1) = v1 + v2 e5 o

  10. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 (e2) = v2 + v3 e5 o

  11. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 (e3) = v4 + v5 e5 o

  12. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 (e4) = v5 + v6 e5 o

  13. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 (e5) = v4 + v6 e5 o

  14. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 : C1  C0 Extend linearly: v4 v6 v1 v3 (e3) = v4 + v5 (e4) = v5 + v6 (e1) = v1 + v2 (e5) = v4 + v6 e5 (e2) = v2 + v3 (Sniei) = niS (ei) o o o o o o o o

  15. =

  16. B0 = Image of = column space of o1

  17. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  18. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  19. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  20. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  21. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 H0 = Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> H0 = Z0/B0 = <[v1], [v4]> where [v1] = {v1, v2, v3} and [v4] = {v4, v5, v6} e5

  22. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  23. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 Using arbitrary coefficients

  24. v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 Using arbitrary coefficients

  25. Row operations v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  26. Row operations v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  27. Row operations using arbitrary coefficients v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  28. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 H0 = Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> H0 = Z0/B0 = <[v1], [v4]> where [v1] = {v1, v2, v3} and [v4] = {v4, v5, v6} e5

  29. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 C1  C0  0 Z0 = kernel of = {x : (x) = 0} = C0 = Z2[v1, v2, v3, v4, v5, v6] e5 o1 o0 o0 o0

  30. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 C1  C0  0 Z0 = kernel of = null space of M0 = = {x : (x) = 0} = C0 = Z2[v1, v2, v3, v4, v5, v6] e5 o1 o0 o0 o0

  31. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 C1  C0  0 Z0 = kernel of = null space of M0 = [0 0 0 0 0 0] = {x : (x) = 0} = C0 = Z2[v1, v2, v3, v4, v5, v6] e5 o1 o0 o0 o0

  32. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 H0 = Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> H0 = Z0/B0 = <[v1], [v4]> where [v1] = {v1, v2, v3} and [v4] = {v4, v5, v6} e5

  33. = C1  C0  0 H0 = Z0/B0 = (kernel of )/ (image of ) null space of M0 column space of M1 Rank H0 = Rank Z0– Rank B0 Z0= null space of [0 0 0 0 0 0] B0 = column space of o0 o1 o1 o0

  34. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 H0 = Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> H0 = Z0/B0 = <[v1], [v4]> where [v1] = {v1, v2, v3} and [v4] = {v4, v5, v6} e5

  35. Cn+1  Cn Cn-1 . . . C2 C1  C0  0 Hn = Zn/Bn = (kernel of )/ (image of ) null space of Mn column space of Mn+1 Rank Hn = Rank Zn – Rank Bn = on+1 on o1 o0 o2 on on+1

  36. C2  C1  C0 H1 = Z1/B1 = (kernel of )/ (image of ) null space of M1 column space of M2 Rank H1 = Rank Z1 – Rank B1 = o1 o2 o1 o2

  37. C2  C1  C0 Z1 = kernel of = null space of M1 o1 o1 o2

  38. C2  C1  C0 Z1 = kernel of = null space of M1 o1 o1 o2

  39. C2  C1  C0 Z1 = kernel of = null space of M1 o1 o1 o2

  40. C2  C1  C0 Z1 = kernel of = null space of M1 = <e3 + e4 + e5> o1 o1 o2

  41. C2  C1  C0 v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 o1 o2 o2 B1 = image of = column space of M2

  42. C2  C1  C0 v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 o1 o2 o2 B1 = image of = column space of M2 = <{v4, v5} + {v5, v6} + {v4, v6}> = <e3 + e4 + e5>

  43. C2  C1  C0 v5 v2 e3 e4 e1 e2 = v4 v6 v1 v3 H1 = Z1/B1 = (kernel of )/ (image of ) null space of M1 column space of M2 <e3 + e4 + e5> <e3 + e4 + e5> Rank H1 = Rank Z1 – Rank B1 = 1 – 1 = 0 e5 = o2 o1 o2 o1

  44. C2  C1  C0 v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 o1 o2 o2 B1 = image of = column space of M2 = <{v4, v5} + {v5, v6} + {v4, v6}> = <e3 + e4 + e5>

  45. C2  C1  C0 v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 o1 o2 o2 B1 = image of = column space of M2 = <{v4, v5} + {v5, v6} + {v4, v6}> = <e3 + e4 + e5>

  46. C2  C1  C0 v5 v2 e3 e4 e1 e2 = v4 v6 v1 v3 H1 = Z1/B1 = (kernel of )/ (image of ) null space of M1 column space of M2 <e3 + e4 + e5> <e3 + e4 + e5> Rank H1 = Rank Z1 – Rank B1 = 1 – 1 = 0 e5 = o2 o1 o2 o1

  47. http://www.ima.umn.edu/2008-2009/ND6.15-26.09 http://www.ima.umn.edu/2008-2009/ND6.15-26.09/abstracts.html#8322

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