AMAT 842, Section 2: Persistent Homology
Spring 2025 (Class #9560)
Monday, Wednesday 4:30-5:50
Room: Humanities 137
Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office: Hudson 137
Office Hours: M-W 11:00-12:00; additional office hours by appointment
Course Notes
About this Course:
This is a Ph.D. level course on persistent homology, the main technical tool of topological data analysis. The focus of the course will be on mathematical foundations. I have taught a similar course twice before; the second time was a two-semester version focusing on multiparameter persistence. During these previous iterations, I prepared and refined a set of typed course notes, which we will also follow this semester. In a single semester, will only be able to cover part of the notes, and so there will necessarily be less emphasis on multi-parameter persistence, but I will cover as much multiparameter material as possible.
Here is a list of possible topics:
- Fundamentals of 1-parameter persistence: Filtrations on point cloud and metric data, the structure theorem for persistence modules, the persistence algorithm
- Interleavings and stability theory
- Construction of multiparameter filtrations from data
- The difficulty of defining (unsigned) barcodes in the multiparameter setting
- Elements of quiver representation theory
- Signed barcodes
- Minimal presentations/resolutions and their computation
- Computable metrics on multiparameter persistence modules
- Computation of density-sensitive bifiltrations
- Visualization of 2-parameter persistent homology
- Vectorization of persistence modules
- Applications
Course Materials:
The main course text will be my typed notes (see the link above). the notes draw in part on some material from my recent article with Magnus Botnan, "An Introduction to Multiparameter Persistence," which may also be a helpful resource.
Prerequisites:
This course requires familiarity
with homology theory and the basics of abstract algebra (groups,
rings), or willingness to learn aspects of this quickly. 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.
Grading and Coursework:
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. There will also be assigned readings from the course notes. For a grade higher than a B, students will be required to do homework (including one brief presentation on a TDA application), and to take a midterm and final.
Academic Regulations:
Naturally, the University's Standards of Academic Integrity apply to
this course, and students are expected to be familiar with these.
Additional TDA Resources: