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Mirror symmetry is a conjectural duality between
symplectic and algebraic geometry. In particular, mirror symmetry
assigns to many integrable systems a complex manifold with potential
function; recent adiabatic techniques have made the computation of
these potentials much more accessible. I will talk about how these
computations look for integrable systems arising hundreds of years
ago, for example spherical pendulum; the cubic surface or the
Grassmannian. In the last few minutes I will try to mention some
recent work (joint with Xiao) on the categorical version of mirror
symmetry for Grassmannians.