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

3. Outline My Topological Emphasis History (3 Perspectives) Equivalence classes as broad mathematics Specific emphasis upon geometric design

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

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

6. Regular Closed Sets: Closed Algebra Boolean Operations A. Requicha

7. Non-manifold Topology K. J. Weiler (Mixed Dimensional (N. F. Stewart))

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

9. KnotPlot !

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

11. Unknot

12. Bad Approximation! Self-intersect?

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

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

15. Another View Approximation as operation in geometric design Need for equivalence Equivalence classes: –Algebra: homorphisms & groups –General topology: homeomorphisms & spaces –Knot theory: isotopies & knots

16. 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

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

19. UMass, RasMol

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

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

23. Global separation

25. Tubular Neighborhoods Depends upon estimates of these values. Similar to medial axis, but avoids its explicit construction, which can be unstable!! and Ambient Isotopy

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

39. 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)

40. Credits ROTATING IMMORTALITY –www.bangor.ac.uk/cpm/sculmath/movimm.htmwww.bangor.ac.uk/cpm/sculmath/movimm.htm KnotPlot –www.knotplot.com

41. 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/parval b.jpg

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