Computational Algebraic Topology

Welcome to Computational Algebraic Topology! Lecture notes for all 8 Weeks can be found under the Lectures tab below. And you can also download a single PDF containing the latest versions of all eight chapters here. See also this playlist containing pre-recorded lecture videos.

The first part of this course, spanning Weeks 1-5, will be an introduction to fundamentals of algebraic topology. The second part of this course, spanning weeks 5-8, will center around material pertaining to topological data analysis.

Explicitly, here is the course syllabus: simplicial complexes, geometric realisations and simplicial maps; homotopy equivalence, carriers, nerves and fibres; homology and its computation; exact sequences and the snake lemma; cohomology, cup and cap products, poincare duality; persistent homology; cellular sheaves and their cohomology; discrete Morse theory.

Important Notice:The course is graded via a final miniproject which covers the above material.

Here is the latest version of the lecture notes (Weeks 1-8) in a single pdf. Here is a YouTube playlist containing pre-recorded videos of this course from a few years ago.

Here are the notes and videos organised by weeks:

Week 1: Complexes notes and videos
Week 2: Homotopy notes and videos
Week 3: Homology notes and videos
Week 4: Sequences notes and videos
Week 5: Cohomology notes and videos
Week 6: Persistence notes and videos
Week 7: Sheaves notes and videos
Week 8: Gradients notes and videos

Hand-written lecture notes from 2020 are under the Antiques tab.

Here are the problem sheets for the course. If you are seeking more thrills, consult the Exercises at the end of every Chapter in the Lecture notes.

Problem Sheet One.
Problem Sheet Two.
Problem Sheet Three.
Problem Sheet Four.

Here are the (hand-written, scanned) lecture notes from Hilary 2020.

Lec 01: Data and Simplicial Complexes
Lec 02: Simplicial Homotopy and Nerves
Lec 03: Chain Complexes and Homology
Lec 04: Functoriality and Exact Sequences
Lec 05: Cohomology and Poincare Duality
Lec 06: Persistence and Stability
Lec 07: Cellular sheaves and cohomology
Lec 08: Discrete Morse theory