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Multidimensional Persistence: Applications in Scientific Engineering & Algebraic Topology

Explore the theory of multidimensional persistence and its applications in scientific and engineering fields through algebraic topology. This Fall 2013 course offered by the University of Iowa covers topics like 1D persistence modules and computing multidimensional persistence using Groebner bases and algorithms. Learn about functoriality, diagrams, and challenges in classifying diagrams in this specialized field. Resources such as Gunnar Carlsson's research and the Dionysus software are invaluable for studying and implementing multidimensional persistence concepts.

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Multidimensional Persistence: Applications in Scientific Engineering & Algebraic Topology

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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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