Bethe Ansatz is the art of diagonalization of certain distinguished commutative subalgebras in quantum affine Lie algebras.
The latter have been realized geometrically in the work of Nakajima, Nekrasov-Shatashvili, Maulik-Okounkov, among others. This provides a geometric solution to the diagonalization problem, which I will explain following our joint paper with Mina Aganagic.