Applied and Computational Topology

Math 689-001 (Spring 2025)

Detailed Syllabus

The following table contains the tentative schedule for the course. It will be updated regularly throughout the semester. The Pages column refers to the pages in the notes that are covered each week. Further book recommendations for additional reading can be found at the end of this page.

Week	Dates	Pages
			I. Introduction
1	01/21 - 01/24	1-12	1. Fixed Points and Homeomorphisms
			2. Algebraic Topology in Applications
2	01/27 - 01/31	13-28	3. Solving a Maze with Topology
			4. Homology of Graphs
			II. Complexes
3	02/03 - 02/07	29-36	1. Simplicial Complexes
			2. Abstract Simplicial Complexes
4	02/10 - 02/14	37-42	3. Simplicial Maps and Approximations
	02/12		No class! (Snow Day)
5	02/17 - 02/21	43-55	4. Homotopy Equivalence
			5. Convex Set Systems and Nerves
6	02/24 - 02/28	56-71	6. Point Clouds and the Cech Complex
			7. Vietoris-Rips Complexes
			8. Delaunay Complexes
			9. Alpha Complexes
			III. Homology
7	03/03 - 03/07	72-80	1. Homology of Chain Complexes
			2. Simplicial Homology
8	03/10 - 03/14		No class! (Spring Break)
9	03/17 - 03/21	81-90	3. The Universal Coefficient Theorem
10	03/24 - 03/28	91-102	4. Cubical Homology
			5. Relative Homology and Excision
11	03/31 - 04/04	103-114	6. Lefschetz Complexes
			7. Chain Maps and Chain Homotopies
12	04/07 - 04/11	115-	8. Homology of Maps and the Degree
			9. Exact Sequences
			IV. Persistence
			1. Topology of Sublevel Sets
			2. Persistent Homology Groups
			3. Persistence Diagrams
			4. Computing Persistence
			5. Stability Theorems
			V. Combinatorial Topological Dynamics
			1. Discrete Morse Theory
			2. Multivector Fields
			3. Conley Theory
			4. Connection Matrices
			5. Application to Classical Dynamics
16	05/05		Student Presentations

In addition, the following books might be useful as secondary reading:

H. Edelsbrunner, J.L. Harer: Computational Topology, American Mathematical Society, 2010. [EH]
T. Kaczynski, K. Mischaikow, and M. Mrozek, Computational Homology, Springer, 2004. [KMM]
K.P. Knudson, Morse Theory: Smooth and Discrete, World Scientific, 2015. [K]
M. Mrozek, T. Wanner, Connection Matrices in Combinatorial Topological Dynamics, Springer, 2025. [MW]
J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984. [M]
N.A. Scoville, Discrete Morse Theory, American Mathematical Society, 2019. [S]

For information on Julia, please see the following books. All of them are available via the library’s ebook subscription.

C. Heitzinger: An Introduction to the Julia Language, Springer, 2022.
N. Kalicharan: Julia - Bit by Bit, Springer, 2021.
A. Lobianco: Julia Quick Syntax Reference, Apress, 2025.