AMAT 583, Section 1: Topological Data Analysis I

Fall 2019, Class #7709

Tuesday, Thursday 2:45-4:05 ES146

Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office: Earth Sciences 120D
Office Hours: Thursday 4:10-5:10, Friday 1:00-2:00, and by appointment
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Lecture Notes

Homework

Exams and practice exams

Exam Dates :
Midterm : Tuesday October 22 (tentative),
Final: Mon. Dec 16 (1:00 - 3:00).

About this Course:
This is the first course in two-semester sequence on Topological Data Analysis (TDA), aimed primarily at students in Albany's Data Science MS program. No prior knowledge of topology is assumed. After a brief introduction to TDA, we will spend the first part of the semester studying basic concepts in topology needed for TDA, including:

topological spaces,
homeomorpism,
homotopy equivalence,
connectednesss,
quotient spaces,
simplicial complexes,
homology (with field coefficients).

As part of our study of homology, we will also devote some time to studying ideas in abstract linear algebra, especially quotients of vector spaces. Later in the semester, we will begin the study of persistent homology, which we will continue in the second semester. If time permits, we will discuss the Mapper algorithm, and connections between persistent homology and clustering, though these topics might have to wait until the spring.

Course Materials:
I plan to post copies of my (handwritten) lecture notes after each class, usually on the same day as the lecture.

The official course text is "Computational Topology" by Edelsbrunner and Harer. I will cover parts of this book later in the semester, when we come to persistent homology. A (free) online version with much of the same content as the published version is here.

Other recommended reading:

Edelsbrunner and Harer's book does not discuss some basic ideas from point-set topology that I wish to cover. For this, 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.
For some of the abstract linear algebra that we will be covering, a nice text is Axler's "Linear Algebra Done Right."

Additional resources include:

The textbook "Persistence Theory: From Quiver Representations to Data Analysis," by Steve Oudot,
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.

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, and this rule will be enforced strictly.

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

35%: Homework and quizzes.
30%: Midterm
35%: Final
2% (bonus): 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.