AMAT 583, Section 4: Topological Data Analysis I

Spring 2021, Class #10398

Monday,Wednesday 4:30-5:50 ES146

Instructor: Michael Lesnick
mlesnick [at] albany [dot] [the usual thing]
Office Hours: T, Th 1:00-2:00, and by appointment.
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Exam Dates :
Midterm : Wednesday March 3 (tentative),
Final: Wednesday May 12, 3:30 - 5:30.

About this Course:
This is the first course in a 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. This is an online-synchonous course, taught on Zoom. Students are expected to attend and partipate, to the extent possible. Recordings of all live classes will be made avaialble. Course materials, including Zoom links for class and office hours, will be hosted on Blackboard.

Topics to be covered (in this course and in TDA II) include:

topological spaces
homeomorpism
homotopy equivalence
path connectednesss
metric spaces
graphs and simplicial complexes
abstract linear algebra (in particular, quotient vector spaces)
homology (with field coefficients)
topological approaches to clustering
persistent homology (definition, computation, stability, applications)
mapper
multiparameter persistent homology

Course Materials:
I plan to post copies of my handwritten lecture notes before each class. These and the course videos are the primary course materials. I may also sometimes provide typed notes.

Recommended reading/viewing:

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 "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.
For point-set topology, good resources include "Introduction to Topology: Pure and Applied" by Adams and Franzosa, and "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."

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. 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.

45%: Homework and quizzes.
20%: Midterm
25%: Final
10%: Class participation / engagement.

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

Pandemic-Related Challenges:
The pandemic--and the move to online instruction in particular--creates a complex set of potential difficulties for students. I intend to hold this class to a high standard of effort, but at the same time, I am mindful of the unique challenges our situation presents, 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 let me know.

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.