The Yang–Mills Millennium problem

Abstract

The Yang–Mills Millennium Prize problem is one of the great challenges of mathematical physics. In the quarter century since it was set, what progress has been made? This Review outlines the problem from a physics point of view, gives its physical background, explains its nature and significance as a problem in mathematics and surveys promising approaches from recent years. Yang–Mills theory is the basis of the standard model of particle physics. The Yang–Mills Millennium Prize problem, to show that the theory is mathematically well defined and that it has the mass gap property, is one of the great challenges of mathematical physics. This Review explores the problem from both physical and mathematical points of view and surveys promising approaches from recent years.