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Computational Topology : A Personal Overview

Computational Topology : A Personal Overview. T. J. Peters www.cse.uconn.edu/~tpeters. My Topological Emphasis:. General Topology (Point-Set Topology) Mappings and Equivalences. Vertex, Edge, Face: Connectivity. Euler Operations.

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Computational Topology : A Personal Overview

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  1. Computational Topology :A Personal Overview T. J. Peters www.cse.uconn.edu/~tpeters

  2. My Topological Emphasis: General Topology (Point-Set Topology) Mappings and Equivalences

  3. Vertex, Edge, Face: Connectivity Euler Operations Thesis: M. Mantyla; “Computational Topology …”, 1983.

  4. Contemporary Influences • Grimm: Manifolds, charts, blending functions • Blackmore: differential sweeps • Kopperman, Herman: Digital topology • Edelsbrunner, Zomordian, Carlsson : Algebraic

  5. KnotPlot !

  6. Comparing Knots • Reduced two to simplest forms • Need for equivalence • Approximation as operation in geometric design

  7. Unknot

  8. Bad Approximation! Self-intersect?

  9. Why Bad? No Intersections! Changes Knot Type Now has 4 Crossings

  10. Good Approximation! Respects Embedding Via Curvature (local) Separation (global) But recognizing unknot in NP (Hass, L, P, 1998)!!

  11. NSF Workshop 1999 for Design • Organized by D. R. Ferguson & R. Farouki • SIAM News: Danger of self-intersections • Crossings not detected by algorithms • Would appear as intersections in projections • Strong criterion for ‘lights-out’ manufacturing

  12. Summary – Key Ideas • Space Curves: intersection versus crossing • Local and global arguments • Knot equivalence via isotopy • Extensions to surfaces

  13. UMass, RasMol

  14. Theorem: If an approximation of F has a unique intersection with each normalof F, then it is ambient isotopic to F. Proof: 1. Local argument with curvature. 2. Global argument for separation. (Similar to flow on normal field.)

  15. Good Approximation! Respects Embedding Via Curvature (local) Separation (global) But recognizing unknot in NP (Hass, L, P, 1998)!!

  16. Global separation

  17. Mathematical Generalizations • Equivalence classes: • Knot theory: isotopies & knots • General topology: homeomorphisms & spaces • Algebra: homorphisms & groups • Manifolds (without boundary or with boundary)

  18. Overview References • Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Denison, planning with Applied General Topology • NSF, Emerging Trends in Computational Topology, 1999, xxx.lanl.gov/abs/cs/9909001 • Open Problems in Topology 2 (problems!!) • I-TANGO,Regular Closed Sets (Top Atlas)

  19. Credits • ROTATING IMMORTALITY • www.bangor.ac.uk/cpm/sculmath/movimm.htm • KnotPlot • www.knotplot.com

  20. Credits • IBM Molecule • http://domino.research.ibm.com/comm/pr.nsf/pages/rscd.bluegene-picaa.html • Protein – Enzyme Complex • http://160.114.99.91/astrojan/protein/pictures/parvalb.jpg

  21. Acknowledgements, NSF • I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS-0138098. • SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR - 0226504. • Computational Topology for Surface Approximation, September 15, 2004, #FMM -0429477.

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