Introduction to Topological Data Analysis
Math 489-500
Spring 2019
Weekly Readings and Assignments
Week #1 The Nature of Data and Data Analysis, January 14 - January 18
Data – Form and Function
Big Data and Some Data Sets
Data Analysis - Graphical, Statistical, Topological
Read:
View:
Do:
- Conduct a web search to find an expository article where "big data" was exploited to solve a problem of importance
in science, engineering,
business, or government and write a brief summary (max half a page) of the type of
data involved and the results. Due Wednesday 1/23/19.
Week #2 Review of Linear Algebra, January 21 - January 25
Vector spaces and subspaces
Examples
Basis and dimension
Linear transformations and matrix representations
Linear Algebra Help:
Week #3 Continued Review of Linear Algebra, January 28 - February 1
Kernel and cokernel of a linear transformation
Quotient vector spaces
Exact sequences and some elementary homological algebra
Inner product spaces
Normed spaces
Introduction to metric spaces
Definition of a metric space
Examples of metric spaces
Metric Space Help:
Do:
- Do the problems listed below from pages 25-30 in Handout #2 from the previous week. They will be due on in class on Monday 2/4/19.
- Problems 1, 2, 5, 6, 7, 17 (this one is a little difficult but worth doing).
Midterm Project Instructions - Due March 18th
Our TA, Zecheng Zhang, will be holding an office hour every Wednesday in Blocker 619 from 2pm to 3pm to assist you in the event
you have programming
questions. My office hours remain Wednesdays 10am to noon and Thursdays 12:40 to 1:30 with other times by appointment.
Week #4 Metric Spaces and Point Clouds, February 4 - February 8
Open and closed sets
Continuous maps between metric spaces
Category of finite metric spaces and non-expansive maps
Injective Envelopes
Some elementary topology
Introduction to Data Clustering
Note: The first project will now be due on Monday March 18th. Instructions for the project will be passed out on Monday February 4th.
Read:
Do:
- Do the problems listed below from pages 53-56 in Handout #2 from week #2. They will be due in class on Monday 2/11/19.
- Problems 6, 8, 9, 15, 19
Week #5 Data Clustering, February 11 - February 15
Clustering Theory versus Practice
Desirable Properties and Kleinberg's Impossibility Theorem
Computationally Practical Methods - K-means Clustering
Hierarchical Clustering / Agglomerative vs. Divisive Clustering
Functorial Clustering a la Carlsson and Memoli - Single Linkage Clustering
Read:
View:
Do:
- Investigate various software packages like R, Mathematica, or Matlab (pick one) for both single-linkage and/or K-means
clustering function calls and use it to cluster
a small data set (maybe 10 points in 2d) that you make up. Then write a brief
description of what you found, what your data set was, and the results of clustering.
For single-linkage the result will be a
hierarchy of clusterings/partitions. This will be due on Wednesday, February 20th.
Midterm Project Instructions - Due March 18th
Our TA, Zecheng Zhang, will be holding an office hour every Wednesday in Blocker 619 from 2pm to 3pm to assist you in the event
you have programming questions. My office hours remain Wednesdays 10am to noon and Thursdays 12:40 to 1:30 with other times by appointment.
Week #6 More Data Clustering and Topology February 18 - February 22
Functorial Overlapping Clustering
Kruskal's Algorithms
Mixture Models and Generalized PCA
Basic Topological Notions
Homeomorphisms
Homotopy Equivalence
Read:
View:
Note: Midterm Exam is Friday March 1.
Week #7 Simplicial Complexes I February 25 - March 1
Motivation
Simplicial Complexes
Data and the Vietoris-Rips Complex
Read:
- Handout #3 on Homology of Simplicial Complexes pages 57-64.
Week #8 Simplicial Complexes II March 4 - March 8
More on Simplicial Complexes
Simplicial Maps
Remarks on Simplicial Approximation
Read:
Midterm Project Instructions - Due March 18th
Spring Break March 11 - March 15
Week #9 Homology of Simplicial Complexes I March 18 - March 22
Return to linear algebra: vector spaces generated by a finite set
Chain complexes of vector spaces and boundary maps
Homology of simplicial complexes: simple examples and calculations
Euler Characteristic
Read:
Do:
- Exercises 1, 2, 5, 6, 7 (figure 3.4 is on page 84) in Handout #3 on page 83. Due 4/5/19.
Week #10 Homology of Simplicial Complexes II March 25 - March 29
Rigorous definition of homology of a simplicial complex
Properties of Homology and a Brief Introduction to Advanced Topics
Introduction to functoriality: inclusion maps induce maps on homology
Computing Homology
Vietoris-Rips Complex
Alpha Complex
Persistence Vector Spaces
Read
Note: The first article listed above should be easy to read given what we've covered so far.
Week #11 Point Clouds, Associated Spaces, and Persistence April 1 - April 5
Motivation: why study point clouds?
Examples of point clouds arising from real-world data
Point clouds as finite metric spaces
More on Single-linkage clustering
Homology of the Vietoris-Rips complex of a point cloud
More on persistence vector spaces: definitions of persistence vector space, linear transformations
Finitely-presented persistence vector spaces
Basic properties of persistence vector spaces
Persistent Homology
Read:
Week #12 Persistent Homology I April 8 - April 12
Classification theorem for finitely-presented persistence vector spaces
Barcodes and persistence diagrams
Persistent homology algorithm
Computational examples
Examples and applications of barcodes
Read:
View:
Do:
- Problems 1, 6, 11 on pages 110 and 111 of the Handout (newer version). Note Problem 6 refers to the Snake Lemma on page 102
of the older version of the Handout. It asks you to prove exactness at coker(f1) in the sequence ... -> ker(f3) -> coker(f1) -> coker(f2) -> ... This assignment will be due in class on Wednesday April 24th.
Final Project Instructions - Due April 30th
April 30th is a Tuesday, but is redefined as Friday. Our TA, Zecheng Zhang, will continue holding an office hour every Wednesday
in Blocker 619 from 2pm to 3pm to assist you in the event you have programming
questions.
My office hours remain Wednesdays 10am to noon and Thursdays 12:40 to 1:30 with other times by appointment.
Week #13 Persistent Homology II April 15 - April 19
Week #14 Survey of Advanced Topics April 23 - April 26
Week #15 Review April 29 and April 30
Note: Final Exam is Friday May 3 from 10:30am to 12:30pm.