Orient BlackSwan

About the BookAbout the AuthorTable of Contents

Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering.

The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.

Herbert Edelsbrunner, Duke University, Durham, NC, and Geomagic, Research Triangle Park, NC, and John L. Harer, Duke University, Durham, NC

Preface
A Computational Geometric Topology
I Graphs
I.1 Connected Components
I.2 Curves in the Plane
I.3 Knots and Links
I.4 Planar Graphs
Exercises

II Surfaces
II.1 2-dimensionalManifolds
II.2 Searching a Triangulation
II.3 Self-intersections
II.4 Surface Simplification
Exercises

III Complexes
III.1 Simplicial Complexes
III.2 Convex Set Systems
III.3 Delaunay Complexes
III.4 Alpha Complexes
Exercises

B Computational Algebraic Topology
IV Homology
IV.1 Homology Groups
IV.2 Matrix Reduction
IV.3 Relative Homology
IV.4 Exact Sequences
Exercises

V Duality
V.1 Cohomology
V.2 Poincar´e Duality
V.3 Intersection Theory
V.4 Alexander Duality
Exercises

VI Morse Functions
VI.1 Generic Smooth Functions
VI.2 Transversality
VI.3 Piecewise Linear Functions
VI.4 Reeb Graphs
Exercises

C Computational Persistent Topology
VII Persistence
VII.1 Persistent Homology
VII.2 Efficient Implementations
VII.3 Extended Persistence
VII.4 Spectral Sequences
Exercises

VIII Stability
VIII.1 1-parameter Families
VIII.2 Stability Theorems
VIII.3 Length of a Curve
VIII.4 Bipartite GraphMatching
Exercises

IX Applications
IX.1 Measures for Gene Expression Data
IX.2 Elevation for Protein Docking
IX.3 Persistence for Image Segmentation
IX.4 Homology for Root Architectures
Exercises

References
Index