# DSCI 475: Topological Data Analysis

## Colorado State University, Spring 2021

**Instructor:** Henry Adams
**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] |

## Software resources

- Tutorial on persistent homology using Ripser (live).
- Tutorial and video on persistent homology using Javaplex.
- See this incomplete list of applied topology software options.