Ideas from theoretical physics have had a profound impact on geometry, topology, and representation theory over the last several decades. An early high point of this interaction was Witten's quantum field theoretic interpretation of the celebrated Donaldson invariants, which in turn opened the door to his discovery of the even-more-celebrated Seiberg-Witten invariants. In this talk, we'll explain how more recently this interaction has made possible dramatic advances in geometric representation theory, with a focus on joint work with Sabin Cautis revealing the structure of the coherent Satake category of a complex Lie group. This is an intricate cousin of the constructible Satake category appearing in the geometric Satake equivalence, a cornerstone of the geometric Langlands program. The coherent Satake category turns out to have rich connections to the Fomin-Zelevinsky theory of cluster algebras, as well as to the representation theory of quantum groups and quiver Hecke algebras. However, while these connections can be stated in purely mathematical terms, their discovery hinged crucially on first understanding how to interpret the coherent Satake category in terms of physics --- in fact, the very same physics (4d N=2 supersymmetric Yang-Mills theory) behind the Donaldson and Seiberg-Witten invariants.