Instructor: Michael Lesnick

mlesnick [at] albany [dot] [the usual thing]

Office: CK 385 (Catskill Building)

Office Hours: M-W 4:30-5:30 (in person and on Zoom by appointment); additional office hours by appointment.

Course Notes

This course is a two-semester sequence on multiparameter persistent homology (MPH), an exciting and very active subfield of TDA with a rich theory and great promise for practical applications. In this course, we will study both the theoretical and computational sides of MPH, with an emphasis on aspects most relevant to the development of MPH as a practical data analysis tool.

The course will be taught in a hybrid format: There will be live lectures at UAlbany, and these will be broadcast via Zoom. The lectures will be recorded and asynchronous participation is possible. Students from outside of UAlbany are welcome to participate in the course remotely. UAlbany students with no scheduling conflict are expected to attend the course in person.

Here is a tentative list of topics (for both semesters):

- Fundamentals of 1-parameter persistent homology, including the persistence algorithm and algebraic stability
- Construction of multiparameter filtrations from data
- The difficulty of defining (unsigned) barcodes for MPH
- Elements of quiver representation theory
- Signed barcodes
- Interleavings and stability theory
- Minimal presentations/resolutions and their computation
- Computable metrics on multiparameter persistence modules
- Computation of density-sensitive bifiltrations
- Upset presentations and the bipersistence of smooth functions
- Sheaf-theoretic viewpoints on generalized persistence
- Visualization of 2-parameter persistent homology
- Vectorization of persistence modules
- Applications

The main resource for the course will be a set of typed notes (see the link above). These will be a revised and expanded version of my notes from a previous (1-semester) version of this course. The updated notes will draw on material from my recent article with Magnus Botnan, "An Introduction to Multiparameter Persistence."

Instructor permission is required. This course requires familiarity with homology theory and the basics of abstract algebra (groups, rings). General mathematical maturity, at the usual level of Ph.D. coursework in mathematics at UAlbany, is also required. Prior exposure to TDA is not required, but would be helpful.

The class will use the university's A-E grading scheme.

To get a B in this course, it will suffice to attend class (in person) regularly. Two lectures can be missed without penalty. For a grade higher than a B, students will be required to do homework, take a midterm (both fall and spring), and complete a final project (spring).

Students not officially registered for the course will not receive a course grade, but can submit homework and take the exams, if they wish. remote students are encouraged to complete a final project in the spring.

Naturally, the University's Standards of Academic Integrity apply to this course, and students are expected to be familiar with these.

- Matthew Wright's animated introduction to Persistent Homology
- Magnus Botnan's TDA course notes
- Robert Ghrist's textbook "Elementary Applied Topology"
- Edelsbrunner and Harer's textbook "Computational Topology." A (free) online version with much of the same content as the published version is here
- Dey and Wang's textbook "Computational Topology for Data Analysis"
- Steve Oudot's textbook "Persistence Theory: From Quiver Representations to Data Analysis", by
- Boissonat, Chazal, and Yvinec's textbook "Geometric and Topological Inference"
- Carlsson and Vejdemo-Johansson's textbook "Topological Data Analysis with Applications"
- Vidit Nanda's Computational Algebraic Topology Course Notes and Videos
- Gunnar Carlsson's survey "Topology and Data"
- Otter et al.'s survey "A Roadmap for the Computation of Persistent Homology"
- TDA software on GitHub