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# Computational Topology : A Personal Overview - PowerPoint PPT Presentation

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

T. J. Peters

www.cse.uconn.edu/~tpeters

General Topology (Point-Set Topology)

Mappings and Equivalences

Connectivity

Euler Operations

Thesis: M. Mantyla; “Computational Topology …”, 1983.

• Grimm: Manifolds, charts, blending functions

• Blackmore: differential sweeps

• Kopperman, Herman: Digital topology

• Edelsbrunner, Zomordian, Carlsson : Algebraic

• Reduced two to simplest forms

• Need for equivalence

• Approximation as operation in geometric design

Approximation!

Self-intersect?

No

Intersections!

Changes

Knot Type

Now has 4

Crossings

Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

But recognizing unknot in NP (Hass, L, P, 1998)!!

• 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

• Space Curves: intersection versus crossing

• Local and global arguments

• Knot equivalence via isotopy

• Extensions to surfaces

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.)

Good with each normal

Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

But recognizing unknot in NP (Hass, L, P, 1998)!!

Global separation with each normal

Mathematical Generalizations with each normal

• Equivalence classes:

• Knot theory: isotopies & knots

• General topology: homeomorphisms & spaces

• Algebra: homorphisms & groups

• Manifolds (without boundary or with boundary)

Overview References with each normal

• 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)

Credits with each normal

• ROTATING IMMORTALITY

• www.bangor.ac.uk/cpm/sculmath/movimm.htm

• KnotPlot

• www.knotplot.com

Credits with each normal

• 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

Acknowledgements, NSF with each normal

• 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.