Report on the status of the Yang - Mills problem

As explained in the official CMI problem description set out by Arthur Jaffe and Edward Witten, YangMills theory is a generalization of Maxwell’s theory of electromagnetism, in which the basic dynamical variable is a connection on a G-bundle over four-dimensional space-time. Its quantum version is the key ingredient in the Standard Model of the elementary particles and their interactions, and a solution to this problem would both put this theory on a firm mathematical footing and demonstrate a key feature of the physics of strong interactions. To illustrate the difficulty of the problem, we might begin by comparing it to the study of classical Yang-Mills theory. Mathematically, this is a system of non-linear partial differential equations, obtained by extremizing the Yang-Mills action,