The Moore-Tachikawa conjecture is a statement at the interface of Lie theory, Hamiltonian geometry, and quantum theory. It posits the existence of a set of Hamiltonian spaces which combine via symplectic reduction in a way analogous to the gluing of surfaces with boundaries. Parts of this conjecture have been solved by Braverman-Nakajima-Finkelberg and Ginzburg-Kazhdan. I will present a new partial resolution using the formalism of shifted symplectic geometry. This formalism was first introduced in derived algebraic geometry as a higher version of symplectic geometry, but it also admits a concrete differential-geometric interpretation using Lie groupoids and differential forms, which I will explain. This interpretation allows us to extend the conjecture in various directions, raising interesting questions in Lie theory and Poisson geometry. This is joint work with Peter Crooks.