I like to play the game where you start with the input of a "physical duality" and wind up with the output of a nontrivial mathematical theorem, usually of the form "these two mathematical widgets that look different are secretly the same".
In particular, I am interested in using dualities arising from "Theory X" (a.k.a. the 6d (2,0) SCFT) to motivate/understand/prove results in the geometric Langlands program (and variants/generalisations thereof).