The quantised McKay correspondence

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Let X be a singular algebraic variety, and Y a resolution of its singularities: a basic (and central) example is given by X being a Kleinian surface singularity, and Y its minimal resolution. An influential conjecture in the literature (the Crepant Resolution Conjecture, or CRC in short), predicts a precise relation between the genus zero Gromov-Witten (GW) theory of X and that of Y through a suitable analytic continuation and (partial) specialisation of their arguments. In the last decade, a host of new tools have been developed for studying these conjectural relations: these tools use mirror symmetry to gain analytical control on the quantum cohomology of orbifolds, and they can be effectively employed to prove and improve the implications of the CRC. The state-of-the-art hinges crucially on the use of Givental-type mirror symmetry theorems for toric orbifolds and complete intersections therein, and there are no general technical tools to build upon a similar proof as for the toric case. In particular, the central case of Kleinian singularities, which are labelled by ADE Dynkin diagrams, are non-toric in type D and E, and for which the CRC becomes a highly refined, quantum version of the classical McKay correspondence, is already out of reach of usual mirror symmetry methods.Problem 1. Prove mirror symmetry and the quantum McKay correspondence for Gorenstein surfacesingularities.Problem 2. Extend this to a higher genus, full-descendent McKay correspondence in full generality.