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
1	01/21 - 01/24	1-	I. Introduction
			1. Fixed Points and Homeomorphisms
			2. Algebraic Topology in Applications
2	01/27 - 01/31		3. Solving a Maze with Topology
			4. Homology of Graphs
3	02/03 - 02/07		II. Complexes
			1. Simplicial Complexes
			2. Convex Set Systems
			3. Delaunay Complexes
			4. Alpha Complexes
			5. Lefschetz Complexes
5	02/17 - 02/21		III. Homology
			1. Chain Complexes
			2. Homology of Chain Complexes
			3. Matrix Reduction
			4. Relative Homology
			5. Exact Sequences
8	03/10 - 03/14		No class! (Spring Break)
9	03/17 - 03/21		IV. Persistence
			1. Persistent Homology
			2. Computing Persistence
			3. Extended Persistence
			4. Stability Theorems
			5. Application to Pattern Formation
12	04/07 - 04/11		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]