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Enumerative geometry of curves in an algebraic variety X is traditionally phrased as computations in cohomology of suitable moduli spaces of curves in X. From many perspectives,including application in mathematical physics, it is interesting and important to promote these computations to K-theory, that is, to compute what may called indices of Dirac operators on these (very singular) moduli spaces, as virtual representation of the group Aut(X). These have truly remarkable properties, including surprising relations for different X that swap the weights of automorphisms for degrees of the curve. In my lectures, I will try to explain this and give a sense of the kind of geometric representation theory that lets one get a handle on such phenomena.