This is the page for the TCC course on singularity categories I am running in autumn 2025. The course will run Wednesdays 1-3pm, in a hybrid format, with lectures in Huxley 145 at Imperial and concurrently streamed online. For practical information, please see the TCC webpage here.
Course description: The singularity category of a ring is a triangulated category which contains lots of interesting geometric and homological information. We'll begin with some introductory material on triangulated categories and commutative algebra. Then we'll see three perspectives on singularity categories - firstly as a quotient of the bounded derived category, secondly as the stable category of maximal Cohen-Macaulay modules, and thirdly as matrix factorisations. We'll see some nontrivial examples coming from Kleinian singularities, and, if time permits, some aspects of the dg theory, with a focus on Hochschild (co)homology. This course will be a mixture of representation theory and algebraic geometry.
Lecture notes: As the lectures progress I will upload the notes here and eventually compile them into a finished document. The lectures will be loosely based off of these notes, which you can take to be a preliminary version of the final notes (beware that, particularly near the end of the course, there may be a significant divergence!).
If you want to get in touch with me, my contact information is available on the homepage.