In this paper, we establish the global C^{2,α} and W^{2,p} regularity for the Monge-Amp`ere equation det D^2u = f subject to boundary condition Du(Ω) = Ω^∗, where Ω and Ω^∗ are bounded convex domains in the Euclidean space R^n with C^{1,1} boundaries, and f is a Ho ̈lder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.

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Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over sl2. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes generalized Dynkin diagrams over any simple Lie algebra g at any level k. This answers a recent comment of Terry Gannon on an old question posed by Victor Kac in 1994.