Henry Adams

Applied and Computational Topology


Universidad de Costa Rica, Summer 2017

Instructors: Henry Adams and Joshua Mirth
Email: henry dot adams at colostate dot edu and mirth at math dot colostate dot edu

Class: June 27 - July 6, 2017

Overview: This class is an introduction to applied and computational topology. We will first give an intuitive introduction to topology, including homotopy equivalent spaces, homology groups, and homotopy groups. We next move to the realm of data analysis: given only a dataset, i.e. a finite sampling from a space, what can we say about the space's shape (which may be reflective of patterns within the data)? The main technique we cover is persistent homology; we describe its theoretical underpinnings, discuss examples of how it has been used on real-life data, and provide coding examples. We also discuss zigzag homology and applications to mobile sensor networks.

Portions of our class will use the tutorial for computing persistent homology with the Javaplex software package.


Date Topic Remark

Tues, June 27, 7-10am An introduction to applied and computational topology (Slides) 4th floor room
Čech and Vietoris-Rips simplicial complexes
The nerve lemma and Latschev's theorem
Stability of persistent homology

Wed, June 28, 9am-12pm Simplicial homology computations in Javaplex 2nd floor computer lab
Exercises 1-2 in the tutorial
Persistent homology and Vietoris-Rips in Javaplex
Exercises 8-11 in the tutorial
Jung's theorem and Čech, Vietoris-Rips interleavings

Fri, June 30, 8am-10am Voronoi diagrams and Delaunay triangulations 4th floor room
Alpha complexes (and exercises)
Evasion paths in mobile sensor networks (Slides)

Mon, July 3, 4-7pm Homology computations with different coefficients 4th floor room
Torus and Klein bottle examples
An algorithm for persistent homology

Wed, July 5, 9am-12pm Witness complexes & examples in Javaplex 2nd floor computer lab
Torus and Klein bottle as identification spaces in Javaplex
Real datasets in Javaplex

Thurs, July 6, 4-6pm Homology of the projective plane 4th floor room
Vietoris-Rips complexes of the circle (Slides)
Metric reconstruction via optimal transport (Notes, Paper)

Other Notes

Notes on Čech and Vietoris-Rips complexes
Mathematica demo on Čech and Vietoris-Rips complexes
Mathematica demo on Vietoris-Rips and witness complexes
Mathematica demo on witness complexes

Installing Javaplex in Matlab

Open Matlab. Inside Matlab, type the following.