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1D persistence

MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 30, 2013: Multidimensional Persistence. Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics

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1D persistence

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  1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 30, 2013: Multidimensional Persistence. 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. 1D persistence V0 = H00 V1 = H01 1 1 0 0 Rank L(0, 0) = 2; Rank L(0, 1) = 1; Rank L(1, 1) = 2 Note Rank determines persistence bars and vice versa

  3. 1D persistence V0 = H00 V1 = H01 1 1 0 0 Rank L(0, 0) = 2; Rank L(0, 1) = 1; Rank L(1, 1) = 2 H00, 0 H00, 1 Ho1, 0 Hki, p = Zki /(Bki+pZki) = L(i, i+p)( Hki) U

  4. 1D persistence c a b ta tb V0 = H00 V1 = H01 ab a -t b t c 0 1 1 0 0 (ab) = t(b - a) H0 Persistence module: F[t] + F[t]/(t) + Σ1F[t] ( Σfntn, f , t Σfntn)

  5. The Theory of Multidimensional Persistence, Gunnar Carlsson, AfraZomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbnerbases, Gunnar Carlsson http://www.ima.umn.edu/videos/?id=862

  6. Computing Multidimensional Persistence, Gunnar Carlsson, GurjeetSingh, and AfraZomorodian

  7. S(1, 1) Z2[x1, x2]/x2

  8. Z2[x1, x2]/x22

  9. Z2[x1, x2]/<x1, x2>

  10. Z2[x1, x2]/x12

  11. Z2[x1, x2]

  12. S(1, 1) Z2[x1, x2]/x2 Z2[x1, x2]/x22 Z2[x1, x2]/x22 Z2[x1, x2]/<x1, x2> Z2[x1, x2]/x12 Z2[x1, x2] + + + + +

  13. S(1, 1) Z2[x1, x2]/x2 Z2[x1, x2]/x22 Z2[x1, x2]/x22 Z2[x1, x2]/<x1, x2> Z2[x1, x2]/x12 Z2[x1, x2] In general, use Buchberger’s/ algorithm for constructing a Grobner basis. Algorithm described in Computing Multidimensional Persistence, by Carlsson, Singh, Zomorodian, can be computed in polynomial time. + + + + +

  14. S(1, 1) Z2[x1, x2]/x2 Z2[x1, x2]/x22 Z2[x1, x2]/x22 Z2[x1, x2]/<x1, x2> Z2[x1, x2]/x12 Z2[x1, x2] In general, use Dionysus software, http://www.mrzv.org/software/dionysus/ See next lecture, 1 week from today. + + + + +

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