Lecture Courses in Combinatorics

Chris Godsil: Algebraic graph theory and quantum computing

8th January, 2024

This is a 32-lecture course, with each lecture being about 45 minutes, given by Chris Godsil. Note that the 17th lecture was not recorded, but slides are at least available for it. The other 31 lectures are still of interest, but this needs to be known.

This course will provide an introduction to problems in quantum computing that can be studied using tools from algebraic graph theory. The quantum topics will relate to quantum walks and to quantum homomorphisms, automorphisms and colouring. The tools from algebraic graph theory include graphs automorphisms and homomorphisms, spectral decomposition and generating functions.

Prerequisites: I will assume a solid background in linear algebra and knowledge of what a permutation group is. Other topics will be covered in class, or in the notes. I will assume the knowledge of physics I had when I started on this topic, that is, no knowledge.

Spencer Unger and Assaf Rinot: Set theory, algebra and analysis

9th January, 2023

This is a 23-lecture course, with each lecture being 75 minutes, given by Spencer Unger and Assaf Rinot.

This course will present a rigorous study of advanced set-theoretic methods including forcing, large cardinals, and methods of infinite combinatorics and Ramsey theory. An emphasis will be placed on their applications in algebra, topology, and real and functional analysis.

Tim Gowers: Topics in Combinatorics

8th October, 2020

This is a 28-lecture course with each lecture lasting about 30 minutes.

This course will cover a miscellaneous collection of topics in combinatorics and closely related fields. What the topics have in common is that they all involve proofs that at one time surprised experts by their simplicity. Sometimes these were the first proofs of long-standing open problems, and sometimes they were new proofs of results that had previously been established by much longer arguments. Several of these arguments use ideas and techniques that have gone on to be used by many other people.

Another theme of the course is the sheer diversity of methods that are used in combinatorics. We shall see uses of probability, linear algebra, linear analysis, topology, entropy, multivariate polynomials, tensor rank, concentration of measure, and more. (There will also be one or two arguments that are completely elementary.)