AMAT 840, Section 2: Topics in Topology
Spring 2019, Class #8333
Tuesday, Thursday 2:45-4:05 ES 153
Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office: Earth Sciences 120D
Office Hours: Tuesday 4:05-5:05, and by appointment
Course Notes (new version, in progress, 2022)
Course Notes (old version, 2019)
About this Course:
This course will focus on multiparameter persistent
homology. This is currently a
very active research topic in applied topology, and there has been considerable
recent progress, but much fundamental work remains to be done. We
will focus on the computational aspects and on aspects most
relevant to the use of multiparameter persistence as a practical tool
for data analysis.
The precise topics that we will cover will depend in part on the interests of the
students, but may include:
- A review of 1-parameter persistent homology,
- How multiparameter persistence arises in TDA, and why it is considered important,
- The unavailablity of a good
barcode for multiparameter persistence modules,
- elements of quiver
representation theory,
- interleavings and stability theory,
- Minimal resolutions and Betti numbers,
- The RIVET approach to interactive visualization of 2-parameter
persistent homology,
- Statistical aspects of 2-parameter persistence.
We may also devote some time to topics in 1-parameter
persistence that are especially relevant to the multiparameter
setting, but where multiparameter extensions have not yet been developed.
Recommended Course Materials:
There is no course text. Useful resources include the text "Graded Syzgies" by Peeva; the
text "Persistence Theory: From Quiver Representations to Data
Analysis" by Oudot. For many topics we will discuss, the best
resource is a research article, and I will suggest some readings as
the course progresses.
Prerequisites:
Instructor permission is required. This course is suitable only for students who are comfortable
with homology theory and the basics of abstract algebra (groups,
rings, modules). Parts of the course may require comfort with basic
probability theory. General mathematical maturity, consistent with what
is usually required of a Ph.D. level course-work in mathematics, is also necessary.
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 regularly. Students
may miss up to two classes without penalty.
For a grade higher than a B, students will be required to do homework and complete a final project. The project may
involve implementing algorithms, performing data analysis, writing an expository report, or some combination of these. Projects will be
chosen by the student, in consultation with the instructor. Students may work in pairs or
alone on the project.
There will be no exams.
Academic Regulations:
Naturally, the University's Standards of Academic Integrity apply to
this course, and students are expected to be familiar with these.