The multiplicative universal centralizer $J$ of a complex reductive group $G$ plays an important role in geometric representation theory. For $G$ of ADE type, $J$ is the K-theoretic Coulomb branch of pure gauge theory. I will present joint work with Ben Webster (to appear) establishing two remarkable structures on $J$: a Bruhat stratification and a cluster structure. Time permitting, I will discuss applications of these structures, including a partial resolution of a conjecture of Cautis–Williams, applications to the cohomology of $J$, and the determination of the Donaldson–Thomas transformation on $J$.