Computational Algebraic Topology

Announcements

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.

The first part of this course, spanning Weeks 1-5, will be an introduction to fundamentals of algebraic topology. The second part, spaning Weeks 5-8, will be taught by Prof Samson Abramsky, and will involve an application of sheaf theory to quantum mechanics. Dramatic update: Prof Abramsky is no longer at Oxford. Therefore the second part of this course, spanning weeks 5-8, will center around material pertaining to topological data analysis. Moreover, there will be no Topic (B) for the miniproject, and no quantum mechanics in the course.

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; discrete Morse theory.

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

Lectures

Here is the latest version of the lecture notes (Weeks 1-8) in a single pdf. The weekly notes and videos are available below:


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.

Exercises

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.

Antiques