This is an impossibly broad question, and makes the unwarranted assumption that *Mathematics* is a uniform field. It might make more sense to ask the same question restricted to, say, *Mathematical Logic*, or *Differential Geometry*, or *Number Theory*.
Nevertheless, I think there may be a sense in which certain fields of mathematics
are (temporarily) blocked by major unresolved conjectures, and others are not.
In which case, a breakthrough in a "blocked" field
might advance *Mathematics*—broadly construed—the most.

For example, prior to 2002, I would have ventured: the Poincaré conjecture & Thurston's geometrization conjecture. But since Perelman's resolution, I see (in my limited vision) no equivalent major conjecture outstanding in Differential Geometry. Similarly, one might might view Fermat's last theorem as a key blockage within Number Theory before the 1995 Wiles-Taylor resolution; or twin primes before Zhang's 2013 breakthrough and subsequent polymath's improvements.

My parochial, biased viewpoint is that resolution of the *P=NP* question, ** tomorrow**, would represent
the greatest advance in Mathematics.
But perhaps others think that, e.g., resolution of the Riemann hypothesis would represent the larger advance?
Of course,

*how*a question is resolved makes a huge difference: A narrow proof is less an advance than a broad reconceptualization that reshapes the landscape.

. I welcome responses specific to narrow fields, but also remarks onQMathematics(if such exists!) as a whole—from which we may all learn from one another.

** Editor's note: there is a meta thread about this post at Should we reopen "The resolution of which conjecture/problem would-advance mathematics the most?"**. Please read that before casting open/close votes.