Topological Data Analysis

Math 7970

Summer 2020

CRN, Meeting days/times and credit: CRN 14822, Lectures recorded, and notes will be linked below following lecture.
Office hours via zoom, Wed 12-1. One credit. Grade determined by HW, make a decent effort and you'll get an A.

Course Description: The world is awash in data, and a fundamental problem is to extract meaning from a massive data set. Persistent homology (PH) was introduced by Carlsson-Zomorodian about 20 years ago, as a way of transitioning from a point cloud data set to a topological space (actually, a family of nested topological spaces), and then using tools of algebraic topology to analyze the data set. PH has been used in a wide variety of domains, from understanding activity in the visual cortex to shape analysis to providing new insight into cancer pathology. This month long class (prerequisite: interest in math and undergraduate linear algebra) will introduce students to the basics of PH. The first part will be an Algebra and Topology "Boot Camp", bringing students up to speed on the necessary mathematical background. The second part will outline the passage from point cloud data to topology, and bring us to the forefront of research in the field.

Grading: Your grade will be determined by homework scores. Problems will assigned in class and collected every week.

Academic Integrity: I encourage group work on the homework problems. This does not include copying each others solutions.

Schedule by week:
Lecture 1 (6/02): Algebra basics-rings, ideals, localization and Noetherian rings and Hilbert basis theorem. Notes.
Lecture 2 (6/04): Euclidean domains, and k[t]-modules and linear algebra, rational canonical form. Notes.
Lecture 3 (6/09): Structure theorem for finitely generated Abelian groups, varieties and primary decomposition. Notes.
Lecture 4 (6/11): Topology basics-simplices, simplicial homology, computation of homology groups. Notes.
Lecture 5 (6/16): Persistent homology: going from point cloud data to a filtered simplicial complex. Notes.
Lecture 6 (6/18): Homological algebra, short and long exact sequences, chain homotopy. Notes.
Lecture 7 (6/23): Multiparameter persistent homology: filtrations with more than one parameter. Notes.
Lecture 8 (6/25): More multiparameter persistent homology, derived functors, spectral sequences. Notes.

Lectures 1-3: Artin "Algebra", Chapter 12, or any basic book on algebra.
Lectures 4,6: Schenck Computational Algebraic Geometry, Chapter 5, and Homological algebra basics and
                      Ghrist Homological algebra and data, IAS/Park City Lecture Notes, 25 (2018) p. 273-325.
Lecture 5:      Ghrist Barcodes: the persistent topology of data , Bulletin of the AMS, 45 (2008) p. 61-75 and
                      Weinberger What is...Persistent Homology, Notices of the AMS, 58 (2011) p. 36-39.
Lectures 7,8: Harrington, Otter, Schenck, Tillmann Stratifying multiparameter persistent homology, SIAM J. Applied Algebra & Geometry, 3 (2019) 439-471.

Book Draft:
Draft of "Algebraic Foundations for Applied Topology and Data Analysis"
Updated 9/1/21 (hks).