CSE 788 X.14 Topics in Computational Topology: --- An Algorithmic View

1. CSE 788 X.14Topics in Computational Topology: ---An Algorithmic View Lecture 1: Introduction Instructor: Yusu Wang

2. Spring 2011 Lecture 1: Introduction • What is topology • Why should we be interested in it • What to expect from this course

3. Space and Shape

4. Geometry • All about distances and angles • area, volume, curvatures, etc • Euclidean geometry • Riemannian geometry • Hyperbolic geometry • …

5. Motivating Examples I • Graphics • Texture mapping • Continuous deformation

6. Motivating Examples II • Computer Vision • Clustering • Shape space Courtesy of Carlsson et al, On the local behavior of spaces of natural images

7. Motivating Examples III • Sensor networks: • Hole detection • Routing / load balancing Courtesy of Wang et al. Boundary recognition in sensor networks by topological methods Courtesy of Sarkar et al., Covering space for in-network sensor data storage

8. Motivating Examples IV • Structural biology • Motif identification • Energy landscape [Wolynes et al., Folding and Design 1996]

9. Topology • Detailed geometric information not necessary • May even be harmful • Wish to identify key information, “qualitative” structure • Topology • Connectivity

10. Introduction • In general, topology • Coarser yet essential information • Characterization, feature identification • General, powerful tools for both space and functions defined on a space • Elegant mathematical understanding available • However • Difficult mathematical language

11. This Course • Introduce basics and recent developments in computational topology • From an algorithmic and computational perspective • Goal: • Understand basic language in computational topology • Appreciate the power of topological methods • Potentially apply topological methods to your research • www.cse.ohio-state.edu/~yusu/courses/788

12. References • Computational Topology: An Introduction,  by  H. Edelsbrunner and J. Harer, AMS Press, 2009. • Online course notes by Herbert Edelsbrunner on computational topology • Algebraic Topology,  by  A. Hatcher, Cambridge University Press, 2002. (Online version available) • An Introduction to Morse Theory,  by  Y. Matsumoto, Amer. Math. Soc., Providence, Rhode Island, 2002. • Elements of Algebraic Topology,  by  J. R. Munkres, Perseus, Cambridge, Massachusetts, 1984.

13. Course Format • Grading: • Course note scribing: 40% • Final project / survey: 60% • Some timelines: • Week 2: • Sign up for scribing date • Meet me to explain your background, and your potential interests • Week 4-5: • Choose project / survey topics • Week 10 – 11: • Final presentation / report due

14. Introduction to Topology

15. History • Seven Bridges of Königsberg Euler cycle problem Abstraction of connectivity Topology: “ qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated ”

16. Homeomorphism • Connectivity • Intuitively, two spaces have the same topology if one can continuously deform one to the other without breaking, gluing, and inserting new things closed curve self-intersecting curve Trefoil knot open curve Two spaces with the same topology are homeomorphic

17. Relaxation of Homeomorphism • Homotopy equivalent • Homologous

18. Topological Quantities • Homeomorphism —› homotopy equivalence —› homology • Describe the qualitative structure of input space at different levels • Quantities invariant under them • Topological quantities • This course will give • Definition, intuition, and their computation • Also examples of applications

19. Topics • Basics in Topology • 2-manifolds • Classification • Polygonal schema, universal cover • Homology • Computation • Persistence homology • Morse functions • Critical points • Morse-smale decomposition • Reeb graph (contour tree)