I will begin by reviewing the relationship between Hitchin’s Integrable System and 4d N=2 Supersymmetric
Quantum Field Theories. I will then discuss two classes of deformations of the Hitchin system which correspond,
in the physical context, to relevant and marginal deformations of a conformal theory. The study of relevant
deformations turns out to be related to the theory of sheets in a complex Lie algebra and their classification
leads to a surprising duality between sheets in a Lie algebra and Slodowy slices in the Langlands dual Lie algebra
(work done with J. Distler) . If there is time, I will discuss marginal deformations which are related to studying
the Hitchin system as a family over the moduli space of curves including over nodal curves (ongoing project with
J. Distler and R. Donagi) .