The goal of this course is to study the so-called (generalised) Hitchin integrable systems, which form a broad class encompassing many of those classically known and amenable to study using algebro-geometric methods. Their phase space consists of pairs of a vector bundle on an algebraic curve and a twisted endomorphism of it, and conserved quantities are provided by coefficients of the characteristic polynomial of this endomorphism. In the special case where the twist is by the canonical bundle of the Riemann surface, the phase space is the total space of the cotangent space to the moduli space of vector bundles on the curve, whose dimension is exactly twice the number of independent conserved quantities.
We will begin by studying vector bundles on algebraic curves and their cohomology with a view to understanding their deformations. We will see the Lax formalism for the Hitchin integrable system and the separation of variables provided by the poles of Baker-Akhiezer functions. We will be guided by low dimensional examples where we can obtain explicit expressions, in particular the case of rank two bundles on punctured surfaces of genera zero and one.