Representations of compact Lie groups are of fundamental importance in many branches of mathematics and theoretical physics, going back to the famous work of Hermann Weyl. In recent years, mathematicians have realized these representations using the topology of certain algebraic
varieties. They invented two completely different constructions: one using quiver varieties, the other using geometric Langlands duality. A long-standing open problem is to relate these two geometric constructions. I will explain how we answer this question using the notion of symplectic duality and some ideas from theoretical physics. I will start by introducing and motivating all the notions mentioned above.