Hilbert schemes, separated variables, and D-branes

We explain Sklyanin's separation of variables in geometrical terms and cons truct it for Hitchin and Mukai integrable systems. We construct Hilbert sch emes of points on T*Sigma for Sigma = C, C* or elliptic curve, and on C-2/G amma and show that their complex deformations are integrable systems of Cal ogero-Sutherland-Moser type. We present the hyperkahler quotient constructi ons for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finall y we discuss the connections to physics of D-branes and string duality.