AMAT 584, Sections 2 and 3: Topological Data Analysis II

Fall 2024, Class Numbers 7807, 9254

Section 2: Monday, Wednesday 10:10-11:30 Massry 213

Section 3: Monday, Wednesday 1:10-2:30 Social Science 117

Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office: CK 385 (Catskill Building)
Office Hours: T, Th 1:30-2:30, and by appointment.
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Exam Dates (Tentative) :
Midterm 1: Wednesday, October 9
Midterm 2: Wednesday, November 20
Final (morning class): Tuesday Dec. 17, 1:00-3:00
Final (afternoon class): Monday Dec. 16, 10:30-12:30

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

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

sets and functions
graphs
metric spaces
clustering (with an emphasis on topological approaches)
complexity of algorithms
algorithms for graphs and clustering
Mapper
simplicial complexes
persistent homology
linear and non-linear dimensionality reduction
multiparameter persistence

Course Materials:
Course materials will be hosted on Brightspace.

The main references for TDA II will be my handwritten notes and "Introduction to Algorithms", 4th edition, by Cormen, Leiserson, Stein, Rivest (CLRS). Typed notes may also sometimes be provided.

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 the first part of the class, homeworks will not be handed in. Instead, we will have a short quiz based on the homework, whose date will be announced with the homework assignment.

In the later part of the course, I may stop with the homework-based quizzes and instead have students submit the homework. In this case, homework submitted 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.

If needed to incentivize good attendance, we may also occasionally have short quizzes at the start of class, based on the previous lecture. These will not be announced in advance.

All homeworks/quizzes will be worth the same amount.

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

15%: Homework/Quizzes
25%: Midterm 1
25%: Midterm 2
35%: Final

The lowest two HW/quiz scores will be dropped.
The midterms 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.