Schedule - Winter 2021

This course follows a Tuesday/Thursday Schedule. There is a section for each day, with materials for that day. This schedule is subject to change before a class is held.

Links to readings can be found on the resources page.

Day 00 - 1/12

Introduction to topological data analysis. Recorded Lecture

Reading:

• Introduction of Oudot,
• peruse Chapter 2 of Ghrist,
• peruse “Topological pattern recognition for point cloud data” by Carlsson.

Day 01 - 1/14

Preliminaries: Graphs, Clustering, Disjoint Set/Union-find, the Graph Laplacian. Recorded Lecture

Code:

• union_find.hpp

Reading:

• (Optional) “A tutorial on spectral clustering” by von Luxburg.
• (Optional) You can also find some notes on spectral clustering in Python here.

Day 02 - 1/19

Basic Topology (spaces, maps, homotopy). Simplicial Complexes. Simplicial Maps. Constructions. Recorded Lecture

Reading:

• Ghrist: Preface and Chapter 2.
• (Optional): Munkres Sections 1.1 and 1.2.
• (Optional): Hatcher Chapter 0.

Day 03 - 1/21

Nerve of a cover, witness complexes, Mapper algorithm. Tries. Recorded Lecture

Code: trie.py simplicial_complex.py demo notebook

For an open-source Mapper implementation (in Python), check out Kepler Mapper.

Reading:

• “Topological Estimation Using Witness Complexes” by V. de Silva and G. Carlsson.
• “Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition” by G. Singh, F. Memoli, and G. Carlsson.
• (Optional) “The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes” by J.D. Boissonnat and C. Maria.
• (Optional) “Extracting insights from the shape of complex data using topology” by P. Y. Lum et al.
• (Optional) “Identification of type 2 diabetes subgroups through topological analysis of patient similarity” by L. Li et al.

Day 04 - 1/26

(Cellular) chain complexes, (Cellular) homology, reduction algorithm. Recorded Lecture

Code:

Reading:

• Ghrist Chapter 4
• “Computing Persistent Homology” by A. Zomorodian and G. Carlsson.
• (Optional) “Persistent and Zigzag Homology: A Matrix Factorization Viewpoint” by G. Carlsson, A. Dwarkanath, and B.J. Nelson, sections 2.2-2.5
• (Optional) Hatcher Chapter 2 (2.2 is most relevant)
• (Optional) “Topological Persistence and Simplification” by H. Edelsbrunner, D. Letscher and A. Zomorodian.

Day 05 - 1/28

Filtrations, persistent homology. Recorded Lecture

Code:

Reading:

• Oudot Chapter 2
• “Computing Persistent Homology” by A. Zomorodian and G. Carlsson.
• (Optional) “Persistent and Zigzag Homology: A Matrix Factorization Viewpoint” by G. Carlsson, A. Dwarkanath, and B.J. Nelson, sections 2.6, 2.7
• (Optional) For some nice visualizations, see Anjan Dwaraknath’s Slides: Link

Homework 1

Due noon on Feb. 5, 2021. Link to assignment Demo mapper graph

Day 06 - 2/2

Pairs, barcodes, diagrams, bottleneck distance, features. Recorded Lecture, PDF of iPad notes - note that I also showed some figures from the papers below.

Code:

Reading:

• Oudot Chapter 3, Section 1.1
• (Optional) “Stability of Persistence Diagrams” by S. Cohen-Steiner, H. Edelsbrunner, and J. Harer.
• (Optional) “Gromov-Hausdorff Stable Signatures for Shapesusing Persistence” by F. Chazal, D. Cohen-Steiner, L.J. Guibas, F. Mémoli, and S.Y. Oudot.
• (Optional) “The Ring of Algebraic Functions on Persistence Bar Codes” by A. Adcock, E. Carlsson, and G. Carlsson.
• (Optional) “Statistical Topological Data Analysis using Persistence Landscapes” by P. Bubenik.

Day 07 - 2/4

Quiver Representations, Zigzag Homology. Recorded Lecture, PDF of iPad notes.

Code:

Reading:

• Oudot Chapter 1
• Ghrist Chapters 5.13, 5.15
• (Optional) “Zigzag Persistence” by G. Carlsson and V. de Silva.
• (Optional) “Zigzag Persistent Homology and Real-valued Functions” by G. Carlsson, V. de Silva, and D. Morozov.
• (Optional) “Persistent and Zigzag Homology: A Matrix Factorization Viewpoint” by G. Carlsson, A. Dwarkanath, and B.J. Nelson.

Homework 2

Due noon on Feb. 19, 2021. Link to assignment Link with corrections

Day 08 - 2/9

Interleavings, Interleaving Distance, Isometry Theorem. Recorded Lecture,
PDF of iPad notes.

Reading:

• Oudot Chapter 3
• Ghrist Chapter 10.6
• (Optional) “Metrics for Generalized Persistence Modules” by P. Bubenik, V. de Silva, and J. Scott.

Day 09 - 2/11

Reach, Weak Feature Size, Sampling. Recorded Lecture, PDF of iPad Notes.

Reading:

• Oudot Chapter 4
• (Optional) “Finding the Homology of Submanifolds with High Confidence from Random Samples” by P. Niyogi, S. Smale, S. Weinberger.
• (Optional) “A Sampling Theory for Compact Sets in Euclidean Space” by F. Chazal, D. Cohen-Steiner & A. Lieutier.

Day 10 - 2/16

Klein bottle in Image Patches. Recorded Lecture, PDF of iPad Notes.

Reading:

• Ghrist Chapter 5.14
• Oudot Chapter 5.5
• (Optional) “On the Local Behavior of Spaces of Natural Images” by G. Carlsson, T. Ishkhanov, V. de Silva & A. Zomorodian.
• (Optional) “Topological Estimation Using Witness Complexes” by V. de Silva and G. Carlsson.

Day 11 - 2/18

Outliers. Metric Measure Spaces, Distance-to-Measure. Recorded Lecture, PDF of iPad Notes.

Reading:

• Oudot Chapter 5.6
• (Optional) “Topological consistency via kernel estimation” by O. Bobrowski, S. Mukherjee, J. E. Taylor.
• (Optional) “Geometric Inference for Probability Measures” by F. Chazal, D. Cohen-Steiner & Q. Mérigot.
• (Optional) “Efficient and robust persistent homology for measures” by M. Buchet, F. Chazal, S.Y. Oudot, D.R. Sheehy.

Day 12 - 2/23

Persistent homology optimizations: cohomology algorithm, clearing, compression. Recorded Lecture, PDF of iPad Notes.

Code: BATS reduction

Reading:

• Oudot Chapter 2.2 (especially 2.2.2)
• (Recommended) “A roadmap for the computation of persistent homology” by N. Otter, M.A. Porter, U. Tillmann, P. Grindrod & H.A. Harrington.
• (Optional) “Persistent Cohomology and Circular Coordinates” by V. de Silva, D. Morozov, and M. Vejdemo-Johansson.
• (Optional) “Dualities in persistent (co)homology” by V. de Silva, D. Morozov, and M. Vejdemo-Johansson.
• (Optional) “Persistent Homology Computation with a Twist” by C. Chen and M. Kerber.
• (Optional) “Clear and Compress: Computing Persistent Homology in Chunks” by U. Bauer, M. Kerber, and J. Reininghaus.

Day 13 - 2/25

Homotopy. Discrete Morse theory and simplification. Recorded Lecture, PDF of iPad Notes.

Reading:

• Ghrist Chapter 7, especially 7.8
• (Optional) Hatcher Chapter 2.1 (Homotopy Invariance)
• (Optional) “Morse Theory for Filtrations and Efficient Computation of Persistent Homology” by K. Mischaikow and V. Nanda.

Day 14 - 3/2

Zigzag zoo, sparse filtrations. Recorded Lecture, PDF of iPad Notes.

Reading:

• Oudot 5.3, 5.4, 7.5.2
• (Optional) “Zigzag Zoology: Rips Zigzags for Homology Inference” by S.Y. Oudot, D.R. Sheehy.
• (Optional) “Linear-Size Approximations to the Vietoris–Rips Filtrations” by D.R. Sheehy.

Day 15 - 3/4

Applications to clustering and regularization. Recorded Lecture, PDF of iPad Notes.

Reading:

• Oudot Chapter 6
• (Optional) “Persistence-Based Clustering in Riemannian Manifolds” by F. Chazal, L.J. Guibas, S.Y. Oudot, and P. Škraba.
• (Optional) “A Topological Regularizer for Classifiers via Persistent Homology” by C. Chen, X. Ni, Q. Bai, Y. Wang.
• (Optional) “A Topology Layer for Machine Learning” by R. Brüel-Gabrielsson, B.J. Nelson, A. Dwaraknath, P. Skraba, L.J. Guibas, and G. Carlsson.

Day 16 - 3/9

Multidimensional and Generalized Persistence.

Reading:

• Oudot Chapters 8 & 9
• (Optional) “The Theory of Multidimensional Persistence” by G. Carlsson and A. Zomorodian.
• (Optional) “The Theory of the Interleaving Distance on Multidimensional Persistence Modules” by M. Lesnick.
• (Optional) “Metrics for Generalized Persistence Modules” by P. Bubenik, V. de Silva, and J. Scott.

Day 17 - 3/11

Project presentations.

Reading Period

Reading Period is 3/13-3/15

Finals Period

Final project report will be due in finals period. Date TBD.