AMAT 584, Section 2: Topological Data Analysis II

Spring 2023, Class #6460

Monday,Wednesday 3:00-4:20, Massry 231

Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office: CK 385 (Catskill Building)
Office Hours: M, W 4:30-5:30, and by appointment.

Exam Dates :
Midterm : Wednesday, March 8 (tentative),
Final: Tuesday May 9, 3:30 - 5:30.

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. As I teach it, the first two semesters of the TDA sequence emphasize mathematical foundations and conceptual understanding. The third course in the sequence is a practicum focusing on hands-on work with real data.

This course is a continuation of my TDA I (AMAT 583) course from Fall '22, and will build on material covered in that course. If you have taken AMAT 583 with Boris, I strongly recommend (in most cases) that you also take TDA II with him, as we organize our courses differently.

Topics to be covered (either in TDA I or TDA II) may include:

topological spaces
metric spaces
homeomorphism
homotopy equivalence
path connectedness
clustering (with an emphasis on topological approaches)
graphs and simplicial complexes
persistent homology (definition, computation, stability, applications)
Mapper
multiparameter persistence
linear and non-linear dimensionality reduction

Course Materials:
Course materials will be hosted on Blackboard.

I plan to post copies of my handwritten lecture notes before each class. These are the primary course materials. As a secondary resource, I may also sometimes provide typed notes.

Additional resources:

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"
For point-set topology, good resources include "Introduction to Topology: Pure and Applied" by Adams and Franzosa, and "Topology" by Munkres.
For the abstract linear algebra that is needed to define homology, a nice text is Axler's "Linear Algebra Done Right."

Homework and Quizzes:
Homework will be assigned semi-regularly. In addition, we might occasionally have quizzes. Homeworks and quizzes will be weighted equally. The lowest score among the homework and quizzes will be dropped. Homework is to be handed precisely by the time it is due (you will have a 5 minute grace period), and this rule will be enforced strictly. Homework handed in at most one day late may be accepted with a 30% penalty, 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.

20%: Homework and quizzes.
30%: Midterm
40%: Final
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.