CSE 5559 Computational Topology: Theory, algorithms, and applications to data analysis Lecture 0: Introduction Instructor: Yusu Wang

Lecture 0: Introduction  What is topology  Why should we be interested in it  What to expect from this course

Space and Shape

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

Motivating Examples I  Graphics  Texture mapping  Continuous deformation

Motivating Examples I  Graphics  Texture mapping  Continuous deformation

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

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

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

Topology  Detailed geometric information not sufficient  Or not necessary  Or may even be harmful  Wish to identify key information, “qualitative” structure  Topology  Connectivity

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

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 

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

Course Format  Grading:  Course project / survey:90%  Include every stage of it  Class participation:10%  Some timelines:  Week 3-4:  Meet me to explain your background, and your potential interests  Week 6-7:  Choose project / survey topics  Week 8 – 15:  Final project / survey. Presentation.

Introduction to Topology

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 ”

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

Relaxation of Homeomorphism  Homotopy equivalent  Homologous

Topological Quantities  Homeomorphism —› homotopy equivalence —› homology  Describe the qualitative structure of input space at different levels  Quantities invariant under them (topological quantities) => (Essential) features  Make topologic objects powerful for feature identification and characterization  This course will give  Definition, intuition, and their computation  Also examples of applications

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features Birth time Death time

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

Topological objects  Not just features of space, one can also incorporate geometry, also functions or maps of a space to capture more sophisticated features

To summarize  Topological concepts  Captures coarse yet essential information about data, space, or functions  Topological objects / methods  Are powerful due to their generality  Are flexible, can describe geometric information or any other information of interests (modeled as functions / maps)  They form natural tools for  Feature identification, characterization of space / data

Topics  Basics in Topology  Common complexes  Homology  Persistence  Homology inference from Data  Scalar field analysis  Reeb graph / contour tree  Hierarchical clustering  Mapper and multiscale-mapper  Data sparsification  Issue of noise Or (discrete) Morse theory Will focus on not only concepts, definitions, algorithms, also intuition why they work, and how they can be used.