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

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

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 open curve closed curveself-intersecting curve Trefoil knot 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)