# DSCI 475: Topological Data Analysis

## Colorado State University, Spring 2021

Email: henry dot adams at colostate dot edu

Lectures: TR 2:00-3:15pm Mountain Time on Zoom
Textbook: None required

Overview: Topological techniques for analyzing high-dimensional or complex data. The shape of data may reflect patterns within; e.g. connected components may correspond to groupings, or a circular shape may correspond to periodic behavior. Topics include clustering, dendrograms, a visual introduction to topology, data modeling and visualization, and selected topics from nonlinear dimensionality reduction, graph-based models of data, Reeb graphs, multi-scale approaches to data, and persistent homology.

Syllabus: Here is the course syllabus.

## Videos

Applied Topology 1: Datasets have shape

Applied Topology 2: Topology and homotopy equivalences

Applied Topology 3: A punctured torus is homotopy equivalent to a figure eight

Applied Topology 4: An introduction to the torus and Klein bottle

Applied Topology 5: Spheres in all dimensions

Applied Topology 6: Homology

Applied Topology 7: How do you recover the shape of a dataset?

Applied Topology 8: An introduction to persistent homology

Applied Topology 9: Spaces of 3x3 natural image patches

Applied topology 10: Unsupervised vs supervised learning

Applied topology 11: Clustering and K-means clustering

Applied topology 12: Hierarchical clustering and single-linkage clustering

Applied topology 13: The problem of chaining in single-linkage clustering

Applied topology 14: Čech and Vietoris-Rips simplicial complexes

Applied topology 15: Introduction to a software tutorial for persistent homology and Ripser

Applied topology 16: Sublevelset persistent homology

Applied topology 17: Persistence and local geometry, Part A

Applied topology 18: Persistence and local geometry, Part B

Applied topology 19: Linear dimensionality reduction - Principal Component Analysis (PCA), Part I

Applied topology 20: Linear dimensionality reduction - Principal Component Analysis (PCA), Part II

Applied topology 21: Nonlinear dimensionality reduction - Isomap, Part I

Applied topology 22: Nonlinear dimensionality reduction - Isomap, Part II

Applied topology 23: Paper Introduction: Coordinate-free coverage in sensor networks

Applied topology 24: Evasion paths in mobile sensor networks, Part I

Applied topology 25: Evasion paths in mobile sensor networks, Part II

Applied topology 26: Evasion paths in mobile sensor networks, Part III

Applied topology 27: Evasion paths in mobile sensor networks, Part IV

## Schedule

 Date Class Topic Remark Jan 19 Course overview [Logistics] Jan 21 Topology and data [Slides, Video] Jan 26 A visual introduction to topology and homotopy equivalences [Slides] Jan 28 A visual introduction to homology [Slides] Feb 2 A visual introduction to persistent homology [Slides] Feb 4 Clustering, k-means clustering [Slides] Feb 9 Hierarchical clustering and dendrograms [Slides] Feb 11 Point cloud persistent homology [Slides, Video] Feb 16 Case studies: Point cloud persistent homology [Slides] Feb 18 Sublevelset persistent homology [Slides, Video] Feb 23 Case studies: Sublevelset persistent homology [Slides] Feb 25 Dimensionality reduction: Principal Component Analysis (PCA) [Tutorial, Slides] Mar 2 Nonlinear dimensionality reduction [Slides] Mar 4 Reeb graphs and the mapper algorithm [Video] Mar 9 Coverage problems in sensor networks [Slides] Mar 11 Coverage problems in sensor networks [Slides]