$1 Million Millennium Problem #2
Yang-Mills Theory and the Mass Gap Hypothesis
Yang-Mills Theory should explain why electrons have mass. While Yang-Mills works in practical terms to describe all of the forces of nature other than gravity, it has not yet been worked out as a mathematical theory.
The Yang-Mills Theory is based on equations from quantum physics. Using experimental evidence, scientists are satisfied that the underlying hypothesis is correct. What is missing is the mathematical components starting with the axioms.
See Keith Devlin's "The Millennium Problems: The Seven Greatest Unsolved Puzzles of Our Time, Basic Books, 2002.