The Gaudin model from 3d mixed BF theory

Integrable systems are very special dynamical systems characterised by having a maximal set of integrals of motion. The Gaudin model provides an important and general class of finite-dimensional integrable systems. After reviewing the basic theory of finite-dimensional classical Hamiltonian integrable systems, focusing on the particular example of the classical Gaudin model, I will show how the latter can be obtained from a certain 3 dimensional gauge theory known as 3d mixed BF theory. This construction offers the simplest example of a new paradigm for understanding integrable systems based on gauge theory, originally proposed several years ago by Costello, Witten and Yamazaki. This talk will be mainly based on the work arXiv:2201.07300 and arXiv:2509.05127.