Courses:

In addition to my own courses at Stanford, MPII, and Dartmouth, my monograph has been used by other instructors in their courses:

• Computational Topology and Geometry, Chee Yap, New York University, New York
• Topology for Computing, Otfried Cheong and Sunghee Choi, KAIST, Daejeon, Korea
• Computational Topology (Seminar), Joachim Giesen and Michael Sagraloff, MPII, Saarbrücken, Germany
• Computational Topology, Dimitrina Stavrova, University of Sofia, Bulgaria

Book Description:

The emerging field of computational topology utilizes theory from topology and the power of computing to solve problems in diverse fields. Recent applications include computer graphics, computer-aided design (CAD), and structural biology, all of which involve understanding the intrinsic shape of some real or abstract space. A primary goal of this book is to present basic concepts from topology and Morse theory to enable a non-specialist to grasp and participate in current research in computational topology. The author gives a self-contained presentation of the mathematical concepts from a computer scientist's point of view, combining point set topology, algebraic topology, group theory, differential manifolds, and Morse theory. He also presents some recent advances in the area, including topological persistence and hierarchical Morse complexes. Throughout, the focus is on computational challenges and on presenting algorithms and data structures when appropriate.

This book is based primarily on my doctoral dissertation, available from my papers page. It incorporates material developed for my course on computational topology.

Contents:

• 1. Introduction
• Part I: Mathematics
  • 2. Spaces and Filtrations
  • 3. Group Theory
  • 4. Homology
  • 5. Morse Theory
  • 6. New Results
• Part II: Algorithms
  • 7. The Persistence Algorithms
  • 8. Topological Simplification
  • 9. The Morse-Smale Complex Algorithm
  • 10. The Linking Number Algorithm
• Part III: Applications
  • 11. Software
  • 12. Experiments
  • 13. Applications