The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills discovered that quantum physics reveals a remarkable relationship between the physics of elementary particles and the mathematics of geometric objects. Predictions based on the Yang-Mills equation have been verified in high energy experiments performed at laboratories all over the world: Brookhaven, Stanford, CERN, and Tskuba.
Nonetheless, there are no known solutions to their equations which both describe massive particles and are mathematically rigorous. In particular, the “mass gap” hypothesis, which most physicists take for granted and use in their explanation for the invisibility of "quarks," has never received a mathematically satisfactory justification. Progress on this problem will require the introduction of fundamental new ideas both in physics and in mathematics.
Mathematical Description authored by Arthur Jaffe and Edward Witten
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