Schedule - Winter 2022

Note that the start of classes has been delayed 1 week

This course follows a Monday/Wednesday 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.

Schedule Archives: Winter 2021

Day 00 - 1/10

Introduction to topological data analysis.

Lecture Recording / iPad Notes / Transcribed Notes

Code:

Demo Notebook for Class

Reading:

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

Day 01 - 1/12

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

Lecture Recording / iPad Notes / Transcribed Notes

Code:

Reading:

(Recommended) “An Impossibility Theorem for Clustering” by Jon Kleinberg: link
(Optional) “A tutorial on spectral clustering” by von Luxburg.
(Optional) You can also find some notes on spectral clustering in Python here.

Day 02 - 1/17

MLK Day. No Class

Day 03 - 1/19

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

Lecture Recording / iPad Notes

Code:

Demo Notebook for Class

Reading:

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

Day 04 - 1/24

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

Lecture Recording / Zoom dropped the call for the final 20 minutes, which was a discussion of use of Mapper by Li et al. in the readings below, and a brief overview of the trie data structure.

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.

Homework 1

Due 10:30am on Feb. 9, 2022. Link

Day 05 - 1/26

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

Lecture Recording

Code: chain.ipynb

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 06 - 1/31

Reduction algorithm, persistent homology.

Lecture Recording

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

Day 07 - 2/2

(Persistent) homology wrap-up, basic implementation, clearing & compression optimizations.

Lecture Recording / iPad Notes

Code: filtration.ipynb, reduction.py, chain.py, BATS reduction, Reduction options in BATS

Reading:

“Computing Persistent Homology” by A. Zomorodian and G. Carlsson.
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 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 08 - 2/7

Pairs, barcodes, diagrams, bottleneck distance, features.

Lecture Recording

Code: notebook on diagrams and distances

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 09 - 2/9

Quiver Representations, Zigzag Homology.

Lecture Recording

Code: Algorithm in BATS

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.

Day 10 - 2/14

Interleavings, Interleaving Distance, Isometry Theorem.

Lecture Recording

Reading:

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

Day 11 - 2/16

Interleavings Part II

Lecture Recording

Homework 2

Due 10:30am on March 2, 2022. Link

Day 12 - 2/21

Klein bottle in Image Patches.

Lecture Recording

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 13 - 2/23

Reach, Weak Feature Size, Sampling.

Lecture Recording

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 14 - 2/28

Cohomology, Applications

Lecture Recording

Code:

Reading:

(Recommended) “Hodge Laplacians on Graphs” by L.H. Lim link
(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.

Day 15 - 3/2

Cohomology II - Hodge laplacians, ranking

Lecture Recording

Code:

Reading:

(Optional) “Statistical ranking and combinatorial Hodge theory” by Jiang et al.
(Recommended) “Hodge Laplacians on Graphs” by L.H. Lim link

Day 16 - 3/7

Applications to clustering and regularization.

Lecture Recording

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 17 - 3/9

Homotopy. Discrete Morse theory and simplification.

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 ??

Outliers. Metric Measure Spaces, Distance-to-Measure.

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 ??

Zigzag zoo, sparse filtrations.

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 ??

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 ??

Project presentations.

Reading Period

Reading Period is 3/12-3/14

Finals Period

Final project report will be due in finals period - 3/18 at midnight.