TITLE: Algebraic geometry used "externally" (in problems without obvious algebraic structure).
QUESTION [21 upvotes]: This is a request for a list of examples of problems (or other mathematical situations) that are not initially of algebro-geometric nature, but can be solved or understood by using algebraic geometry.  
Here are some applications that are not of the kind sought:

Diophantine equations or other problems whose basic data are specified in algebraic terms, or have an immediate translation into such terms.
GAGA or reduction to finite characteristic arguments, but applied to problems that are clearly already within (or very near) the algebro-geometric sphere, involving varieties or moduli spaces, or cohomology of such spaces.

REPLY [2 votes]: There are several papers by Michael Atiyah where he studies partial differential equations and distributions by methods of algebraic geometry - especially the Hironaka resolution of singularities. See eg Resolution of Singularities and Division of Distributions or the two articles with Bott and Garding about Lacunas for Hyperbolic Differential Operators with Constant Coefficients.