T. J. Peters Computational Topology : A Personal Overview

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

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

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

Regular Closed Sets: Closed Algebra Boolean Operations A. Requicha

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

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

KnotPlot !

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

Unknot

Bad Approximation! Self-intersect?

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

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

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

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

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

UMass, RasMol

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.

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

Global separation

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

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

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/ Open Problems in Topology 2 (problems!!) I-TANGO,Regular Closed Sets (Top Atlas)

Credits ROTATING IMMORTALITY – KnotPlot –

Credits IBM Molecule – rscd.bluegene-picaa.html Protein – Enzyme Complex – b.jpg

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