Log-epiperimetric inequalities were introduced in seminal work of Reifenberg to study solutions of the Plateau problem, and have since played a fundamental role in understanding the regularity of solutions to various variational problems, including minimal submanifolds and free boundary problems. In this talk, I will introduce this technique and discuss how it can be adapted to the setting of the Yang-Mills functional, a celebrated Lagrangian central to the development of 4-manifolds geometry. Furthermore, I will explain how to establish uniqueness of blow-ups with isolated singularities. This is based on joint work with Riccardo Caniato.