Formalizing Class Field Theory
Date: 21 - 25 July 2025
Location: Mathematical Institute, University of Oxford
Event type: CMI HIMR Summer School
Organisers: Kevin Buzzard (Imperial), Richard Hill (UCL)
Class Field Theory in its cohomological form is one of the highlights of early 20th century mathematics, and is now understood as the abelian case of the Langlands Philosophy. Although it sounds like science fiction to many mathematicians, some computer scientists are arguing that AI methods are progressing so fast that soon computers will be helping humans to push back the boundaries of research in the Langlands Philosophy. However, there is currently no concrete evidence that this is happening. Furthermore, using a language model alone to do mathematics at this level is problematic, because language models are error-prone, and one error in a mathematical argument invalidates it.
This summer school does not have anything to do with AI, but it has a lot to do with class field theory. During the school, we will be teaching class field theory to the Lean theorem prover. You can imagine the school as a group of people collaborating on writing a Bourbaki-like document explaining class field theory. Or you can imagine it as a group of people turning class field theory into a bunch of levels of a puzzle game and then solving these levels. Or you can imagine it as a group of people creating training data for a theorem prover-backed AI which can then try and learn some of these interesting mathematical ideas.
Prerequisites: Students should have a good knowledge of Lean and a basic knowledge of cohomology groups, Cyclotomic fields, Kummer extensions and local fields. It is recommended that students work through some of the free online textbook Mathematics in Lean prior to the summer school.
This event will be taking place in-person. Accommodation, lunch and subsistence will be provided; participants are expected to fund their own travel (limited funds may be available for those unable to secure travel costs from their home institution).
Registration will open in January 2025.
This summer school is jointly funded by the Clay Mathematics Institute, the Heilbronn Institute for Mathematical Research and the Mathematical Institute at the University of Oxford.