CSE 5559
Computational Topology:
Theory, algorithms, and
applications to data analysis
Lecture 0: Introduction
Instructor: Yusu Wang
Lecture 0: Introduction
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What is topology
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Why should we be interested in it
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What to expect from this course
Space and Shape
Geometry
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All about distances and
angles
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area, volume, curvatures, etc
Euclidean geometry
Riemannian geometry
Hyperbolic geometry
…
Motivating Examples I
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Graphics
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Texture mapping
Continuous deformation
Motivating Examples I
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Graphics
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Texture mapping
Continuous deformation
Motivating Examples II
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Computer Vision
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Clustering
Shape space
Courtesy of Carlsson et al, On the local behavior of spaces of natural images
Motivating Examples III
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Sensor networks:
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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
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Structural biology
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Motif identification
Energy landscape
[Wolynes et al., Folding and Design 1996]
Topology
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Detailed geometric information not sufficient
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Or not necessary
Or may even be harmful
Wish to identify key information, “qualitative” structure
Topology
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Connectivity
Introduction
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In general, topology
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Coarser yet essential information
Characterization, feature identification
General, powerful tools for both space and functions defined
on a space
Elegant mathematical understanding available
However
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Difficult mathematical language
This Course
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Introduce basics and recent developments in
computational topology
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Goal:
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From an algorithmic and computational perspective
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/5559
References
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Computational Topology: An Introduction, by H. Edelsbrunner and
J. Harer, AMS Press, 2009.
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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.
Course notes
Course Format
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Grading:
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Course project / survey: 90%
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Include every stage of it
Class participation:
10%
Some timelines:
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Week 3-4:
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Week 6-7:
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Meet me to explain your background, and your potential interests
Choose project / survey topics
Week 8 – 15:
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Final project / survey. Presentation.
Introduction to Topology
History
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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
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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 curve
self-intersecting
curve
Two spaces with the same topology are homeomorphic
Trefoil knot
Relaxation of Homeomorphism
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Homotopy equivalent
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Homologous
Topological Quantities
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Homeomorphism —› homotopy equivalence —› homology
 Describe the qualitative structure of input space at different
levels
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Quantities invariant under them (topological quantities)
=> (Essential) features
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Make topologic objects powerful for feature identification and
characterization
This course will give
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Definition, intuition, and their computation
Also examples of applications
Topological objects
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Not just features of space, one can also incorporate geometry,
also functions or maps of a space to capture more sophisticated
features
Topological objects
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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
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Not just features of space, one can also incorporate geometry,
also functions or maps of a space to capture more sophisticated
features
Death time
Birth time
Topological objects
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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
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Topological concepts
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Topological objects / methods
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Captures coarse yet essential information about data, space, or
functions
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
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Feature identification, characterization of space / data
Topics
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Basics in Topology
Common complexes
Homology
Persistence
Homology inference from Data
Scalar field analysis
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Reeb graph / contour tree
Hierarchical clustering
Mapper and multiscale-mapper
Will focus on not only concepts,
definitions, algorithms, also intuition why
they work, and how they can be used.
Data sparsification
Issue of noise Or (discrete) Morse theory