According to MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural sciences, such as Physics, Chemistry, Biology, Continuum Mechanics, and so on.
What I am interested in, here, is examples where PDEs were used to establish a result in an other mathematical field. Let me provide a few.
- Topology. The Atiyah-Singer index theorem.
- Geometry. Perelman's proof of Poincaré conjecture, following Hamilton's program.
- Real algebraic geometry. Lax's proof of Weyl-like inequalities for hyperbolic polynomial.
Only one example per answer. Please avoid examples in the reverse sense, where an other mathematical field tells something about PDEs (examples: Feynman-Kac formula from probability, multi-solitons from Riemann surfaces). This could be the matter of an other MO question.