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MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 13, 2013: Persistent homology II Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics

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  1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 13, 2013: Persistent homology II 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://link.springer.com/article/10.1007/s00454-002-2885-2

  3. A filtered complex is an increasing sequence of simplicial complexes: C0 C1 C2 … U U U Topological Persistence and Simplification:link.springer.com/article/10.1007/s00454-002-2885-2

  4. s is a positive simplex if it creates a cycle when it enters.. s is a negative simplex if it destroys a cycle when it enters Topological Persistence and Simplification:link.springer.com/article/10.1007/s00454-002-2885-2

  5. Topological Persistence and Simplification:link.springer.com/article/10.1007/s00454-002-2885-2

  6. Topological Persistence and Simplification:link.springer.com/article/10.1007/s00454-002-2885-2

  7. i, p i i+p i Hk = Zk/(BkZk ) U p-persistent kth homology group: http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  8. i, p i i+p i Hk = Zk/(BkZk ) U Topological Persistence and Simplification:link.springer.com/article/10.1007/s00454-002-2885-2

  9. i, p i i+p i Hk = Zk/(BkZk ) U Topological Persistence and Simplification:link.springer.com/article/10.1007/s00454-002-2885-2

  10. http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  11. Computing Persistent Homology by AfraZomorodian, Gunnar Carlsson M1 = http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  12. Computing Persistent Homology by AfraZomorodian,Gunnar Carlsson M1 = http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  13. M1 =

  14. M1 = i, p Hk = Zk/(BkZk ) i i+p i U

  15. Computing Persistent Homology by AfraZomorodian,Gunnar Carlsson i, p Hk = Zk/(BkZk ) i i+p i U http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  16. Let si be a k-simplex. Defn: degree of si = degsi = time when si enters the filtration. . deg a = deg b = 0; deg c = deg d = degab = degbc = 1 http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  17. Let si be a k-simplex. • Defn: degree of si = degsi • = time when si enters the filtration. . • Defn: degtnsi = n + degsi • A polynomial is homogeneous if all terms have the same degree: t3a – t2c + ac is in C3 http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  18. i, p i i+p i Hk = Zk/(BkZk ) U p-persistent kth homology group: http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  19. Computing Persistent Homology by AfraZomorodian,Gunnar Carlsson i, p Hk = Zk/(BkZk ) i i+p i U http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  20. k Ck Ck-1 Let { ej} be a be a homogeneous basis for Ck Let { ei} be a be a homogeneous basis for Ck-1 degei + deg Mk(i, j) = degej i, p Hk = Zk/(BkZk ) i i+p i U

  21. Computing Persistent Homology by AfraZomorodian,Gunnar Carlsson i, p Hk = Zk/(BkZk ) i i+p i U http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  22. Z1 = kernel of 1 = null space of M1 = < ad + cd + t(bc) + t(ab), ac + t2bc + t2ab > B0= image of 1= column space of M1 = < tc + td, tb + c, ta + tb>

  23. H0 = Z0/B0 = (kernel of 0)/ (image of 1) null space of M0 column space of M1 = < a, b, c, d : tc + td, tb+ c, ta + tb> = 0, 0 i, p 2 i i+p i H0 = Z0/(B0Z0 ) H0 = Z2 For p > 0: H0= Z2 U 0, p

  24. < a, b, c, d : tc + td, tb + c, ta + tb> 0, p 0, 0 i, p 2 i i+p i H0 = Z0/(B0Z0 ) H0 = Z2 For p > 0: H0= Z2 U http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  25. < a, b, c, d : tc + td, tb + c, ta + tb> 1, 0 2 H0 = Z2 For p > 0 : H0= Z2 For i> 1 : H0= Z2 i, p i, p i, p i i+p i H0 = Z0/(B0Z0 ) U http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

  26. < a, b, c, d : tc + td, tb + c, ta + tb> [ ) [ ) [ ) [ ) [ http://link.springer.com/article/10.1007%2Fs00454-004-1146-y

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