The Clay Mathematics Institute,Rhode Island US, is offering one million dollars for each of the 7 “Millennium Problems”.
The laws of Quantum Mechanics are fundamental to the world of elementary particles. Yang and Mills postulated a new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory now forms the foundation of the elementary particle theory. Though the predictions of this theory have been tested experimentally, its mathematical foundation is still not clear. Yang Mills theory to describe the strong interactions of elementary particles depends on a quantum mechanical property termed as “mass gap”. The property has not been understood from the theoretical perspective, though it has been discovered by physicists experimentally and through computer simulations. Introduction of new rudimental ideas in mathematics and physics, are required to progress in establishing the existence of Yang Mills theory and “mass gap”.
Here is a more detailed description of the Yang Mills Problem. (PDF)