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: 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.