Symplectic duality, as described by Braden-Proudfoot-Licata-Webster, is an equivalence of certain categories associated to a pair of conical symplectic singularities. Each such category is a subcategory of modules over a deformation quantization of functions on the corresponding singularity. The prototypical example is when the singularity is the nilpotent cone of a semi-simple Lie algebra g in which case the corresponding category is the Bernstein-Gelfand-Gelfand Category O associated to g.

Physicists immediately noticed that all known dual pairs arise as Higgs and Coulomb branches of 3d N = 4 SUSY field theories. However until recently there was no mathematical definition of the Coulomb branch and no physical definition of Category O. In this talk I will survey recent progress by Braverman-Finkelberg-Nakajima and Bullimore-Dimofte-Gaiotto-Hilburn-Kim addressing these questions.