AMAT 584, Section 1: Topological Data Analysis II

Fall 2021, Class #8489

MW 3:00-4:20, Massry Center for Business, B014

Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office Hours: Monday, Wednesday 4:30-5:30, and by appointment
Office Hours Locations: Massry B014, B003, and B125 (see Blackboard for schedule)

Exam Date :
Midterm: Wednesday October 13 (tentative).

About this Course:
This is the second course in a three-semester sequence on Topological Data Analysis (TDA), aimed primarily at students in Albany's Data Science MS program. The course picks up where TDA I (as taught in Spring 2021) left off. Topics will include:

simplicial complexes,
abstract linear algebra and quotient spaces,
homology (with field coefficients),
persistent homology,
multiparameter persistent homology,
applications of TDA (time permitting)

Prequisites:
Students are formally required to have either taken TDA I (AMAT 583) or to have permission of the instructor. In addition, it will be helpful for students to be familiar with the following concepts, which were covered in TDA I in Spring 2021:

metric spaces and topological spaces,
homeomorphism,
homotopy equivalence,
path connectednesss,
equivalence relations,
single linkage clustering,
graphs: connected components and cycles.

Course Materials:
I plan to post copies of my (handwritten) lecture notes before each class, usually on the same day as the lecture. These notes will be the main reference for the course.

For some of the abstract linear algebra that we will be covering, a nice text is Axler's "Linear Algebra Done Right."

For students wishing to brush up on basic point-set topology, a readable resource is "Introduction to Topology: Pure and Applied" by Adams and Franzosa. Another popular text covering point-set topology in more detail is "Topology" by Munkres.

Additional resources:

Matthew Wright's animated introduction to Persistent Homology
Magnus Botnan's TDA course notes,
"Computational Topology" by Edelsbrunner and Harer. A (free) online version with much of the same content as the published version is here
The textbook "Computational Topology for Data Analysis,", by Tamal Dey and Yusu Wang
The textbook "Persistence Theory: From Quiver Representations to Data Analysis,", by Steve Oudot
The textbook "Geometric and Topological Inference", by Boissonat, Chazal, and Yvinec
Vidit Nanda's Computational Algebraic Topology Course Notes and Videos
M ycourse notes on multiparameter persistent homology
The survey "Topology and Data", by Gunnar Carlsson
The survey "A Roadmap for the Computation of Persistent Homology", by Nina Otter et al. This provides a brief overview of persistent homology, discusses software for persistent homology computation, and benchmarks several software packages.
Some Topological Data Analysis Software on GitHub.

Homework and Quizzes:
Homework will be assigned semi-regularly. In addition, we may occasionally have quizzes. Homeworks and quizzes will be weighted equally. The lowest two scores among the homework and quizzes will be dropped. Homework is to be handed in at the beginning of class on the day it is due (you will have a 5 minute grace period), and this rule will be enforced strictly. Tentatively, homework handed in at most one day late may be accepted with a 30% penalaty, or at most two days late with a 50% penalty. You may discuss homework with your classmates, but homework must be written up on your own.

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

40%: Homework and quizzes.
25%: Midterm
25%: Final Project
10%: Class participation / engagement.

NOTE: The midterm and final may be curved, but not downward.

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

There will be no leeway on missed exams or last-minute exam rescheduling, except as noted in the regulations. If you anticipate an issue with the timing of an exam, please let me know as soon as possible.

Pandemic-Related Challenges:
The pandemic creates a complex and ever-evolving set of difficulties for students. I am mindful of this, and I intend to conduct the class accordingly. If you are dealing with issues created or exacerbated by the pandemic that risk getting in the way of your being a focused, active participant in this class, please feel free to discuss your situation with me.