Gaudin models were historically introduced as integrable spin systems, based on the Lie algebra su(2) or more generally any finite-dimensional simple Lie algebra. This talk is devoted to the so-called affine Gaudin models, which are variants associated with infinite-dimensional affine Lie algebras instead of finite ones. As such, they possess an infinite number of degrees of freedom and more precisely take the form of quantum field theories. They are conjecturally integrable, in the sense of admiting an infinite number of commuting conserved operators. In this talk, I will review preliminary results and conjectures on the construction of these operators as well as their diagonalisation, guided in particular by a conjectural affinisation of the Geometric Langlands correspondence. This is based on collaborations with G. Koutousov, A. Molines, J. Teschner, B. Vicedo and C.A.S. Young..

https://indico.math.cnrs.fr/event/15480/