AMAT 483, Section 1: Geometric and Topological Data Analysis

Spring 2025, Class # 9594

Tuesday, Thursday, 1:30-2:50, Massry 007

Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office: Hudson 137
Office Hours: M-W 11:00-12:00; additional office hours by appointment

Exam Dates (Tentative):
Midterm 1: Thursday, March 6
Midterm 2: Wednesday, April 17
Final: Wednesday, May 14, 8:00 a.m.

About this Course:
This is a course about classical and modern methods for studying the shape of data. Data sets often have interesting geometric structure (shape), which can encode useful information. This course will introduce classical and modern methods for probing the shape of data, along with the mathematics underlying these methods. The course will primarily emphasize mathematical foundations and conceptual understanding, but we will also discuss real-world examples and have opportunities for hands-on work with data.

Topics to be covered may include:

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

Course Materials:
Course materials will be hosted on Brightspace.

The primary course reference will be a set of typed course notes by Håvard Bjerkevik, Barbara, Giunti, and me. These notes were originally developed for our Master's-level TDA course, and will be adapted to fit this course. I may also sometimes share handwritten 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:
Homework will be assigned semi-regularly. All homeworks will be weighted equally. 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.

Quizzes:
We may occasionally have short quizzes, which will be worth the same amount as one homework.

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

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

The lowest two HW/quiz scores score will be dropped.
NOTE: The midterm and final may be curved, but not downward.

Attendance is required. You may miss up to two classes without penalty. After that, there will be a penalty of 1% on the final grade per missed class.

Each student will choose between taking a final and doing a final project exploring the shape of real-world data. The final project will require both a presentation in front of the class and a 5-10 page written submission. For most students, taking the final will be the easier option, but the project could be a good choice for students looking to build a data analysis portfolio.

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.