Mirror symmetry and the quantum McKay correspondence

In this thesis, we take a series of steps towards a proof of the quantum McKay correspondence in full generality. We establish a uniform Lie-theoretic mirror theorem for ADE singularities, also covering the as-of-yet untreated non-toric Dynkin series $\mathrm{D}_n$ and $\mathrm{E}_n$, in terms of a suitable one-dimensional Landau--Ginzburg model. We then leverage this to the as-of-yet untreated study of the global Dubrovin connection for the $\mathrm{D}$-series, verifying Iritani's structure conjecture for the CRC in that context. We also push the CRC beyond its original geometric incarnation to encompass the representation-theoretically meaningful case of ``foldings''.