TITLE: In what ways has physics spurred the invention of new mathematical tools?
QUESTION [15 upvotes]: I came across this comment:

Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more interesting: their lack of rigor is usually a reflection of the inadequacy of existing mathematical tools for their purposes. When such a theory is read by a mathematician, he is forced to invent new tools or use existing tools in a way he had never thought to before in order to make sense of it. This in turn has spurred many mathematical advances and even created new fields of study. For example, many of the most important PDEs (heat, wave, Laplace, Navier-Stokes, Schrodinger, etc.) were first encountered in physics. 

(src: http://www.quora.com/Why-does-the-Feynman-path-integral-make-accurate-predictions-in-physics-even-though-it-is-not-rigorously-defined-mathematically/answer/Simon-Segert)
I'm curious in what other ways physics (or other pursuits of physical/scientific/computational truth) have sparked the creation of new mathematical tools to understand and formalize these things. Is one example Turing's formalization of computation, or Shannon's formalization of information?

REPLY [2 votes]: Gauge theory is one example where one sees physics stimulating the development of new mathematics. 15 years ago Seiberg and Witten based on intuitions coming from physics proposed a new powerful way of doing gauge theory and in particular almost trivializing earlier impressive results of Fields medal caliber, such as Donaldson's diagonalisation result for 4-manifolds with definite intersection forms; see this article for a discussion.