** 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.

**References:**

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).