What does Yang-Mills and mass gap problem has to do with mathematics?

There is a long, long list of mathematical subjects that were either pioneered or significantly inspired by results in quantum field theory. However, while physicists may trust the manipulations they do in QFT, and the results of those manipulations have been spectacularly successful, for almost every interesting quantum field theory, there isn't even a rigorous definition or existence proof, much less a justification behind the manipulations that led to the invention of, for example, Seiberg-Witten invariants.

Solving the mass gap problem in Yang-Mills would represent the successful rigorous existence of a very non-trivial quantum field theory and the demonstration of a very nontrivial result about that field theory (that hasn't even been adequately demonstrated using physical techniques). While there probably aren't many direct mathematical consequences to the existence of a mass gap, the techniques involved would almost assuredly be applicable towards the large number of QFT inspired results in mathematics.

Mathematical physics is the study of physical questions from the point of view of full mathematical rigor. Physical questions are phrased as well-defined mathematical problems, to be attacked with methods from differential geometry, functional analysis, Lie groups, topology, etc..

The mathematically rigorous construction of interacting 4-dimensional relativistic quantum field theories is one of the hardest problems of mathematical physics, unsolved since over 50 years, in spite of the continuous efforts of many excellent people. The Yang-Mills quantum field theories are believed to be the most benign ones to be constructed, but the current functional analytic tools were so far not sufficient to produce the bounds needed to show the existence of the appropriate limits.

The techniques developed for an affirmative answer of the existence problem for a Yang-Mills quantum field theory (in the sense required by the Clay Millennium problem) would open a door to a constructive mathematical approach to many other 4-dimensional relativistic quantum field theories, including those describing realistic elementary particle physics.

Stochastic partial differential equations in 4 dimensions (e.g., fluid flow) pose closely related problems, connected through analytic continuation to imaginary time. Thus the new techniques would also find applications there.

For a synopsis of what in particular needs to be achieved see my review http://physicsoverflow.org/21846 of a failed attempt to solve the problem.