Algebraic geometry used "externally" (in problems without obvious algebraic structure). 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. 
 A: The polynomial method is a powerful, albeit somewhat mysterious and fragile, tool in extremal combinatorics, being used for instance in Dvir's proof of the Kakeya conjecture over finite fields:
http://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/
This is currently the only known proof of the full conjecture.  Previously to Dvir's work, algebraic geometry methods did not feature prominently in the prior partial results.
A related method is Stepanov's method to count points in algebraic varieties over finite fields, though this is clearly a question which was already well within the purview of algebraic geometry to begin with.
A: Define the Ramanujan $\tau$-function from $\mathbb N \rightarrow \mathbb Z$ as the Fourier coefficients of the $\Delta$ function; i.e.,
$$ \sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} .$$
This is a pure number-theoretic function. Now the Ramanujan conjecture says that 
$$|\tau(p)| \leq 2p^{11/2} $$
for every prime $p$, which is also a purely number theoretic statement.
Pierre Deligne proved it as a consequence of the Weil conjectures. 
A: 
The construction of certain Steiner systems is a good example.
A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (called blocks) of an $r$-element set $S$ such that every $p$-element subset of $S$ is contained in a unique element of $A$.  Good examples come from considering as blocks the set of hyperplanes in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field.  For example, $\mathbb{A}^2$ over $\mathbb{F}_3$ gives a $(2, 3, 9)$ Steiner system:  it contains $9$ ($\mathbb{F}_3$-rational) points, and let the blocks be the lines, each of which consists of $3$ points.  Then any $2$ points are contained in a unique line.  This is the unique $(2, 3, 9)$ Steiner system.

A: There exists a unique map from permutations of 1,2,3... that move finitely many numbers, to their "Schubert polynomials" in ${\mathbb Z}[x_1,x_2,...]$, satisfying the following recursion: $S_{id} = 1$, and if $w(i) > w(i+1)$, then $S_{w r_i} = (S_w - r_i \cdot S_w) / (x_i - x_{i+1})$. (Here $r_i$ switches $i$ and $i+1$, or $x_i$ and $x_{i+1}$.)
It's not too hard to prove that these are polynomials, form a basis of the polynomial ring, have positive coefficients, and much else. The nonobvious theorem is that the structure constants (expanding a product of two basis elements in the basis) are positive. The only proofs known of this are geometric.
(This is perhaps a lame example, in that the motivation for Schubert polynomials was geometric -- they represent the classes of Schubert varieties in the cohomology rings of flag manifolds.)
A: I think this is an example:
In Arnold's intro PDE book, he discusses Maxwell's theorem 
about spherical functions being expressible in terms of
derivatives of 1/r. The appendix gives topological and
algebraic geometry interpretations.
Unfortunately I don't know enough about it to give a better
description.
A: A really nice example is the (unpublished) work of Larsen and Pink on the "rough" classification of subgroups of $\mbox{GL}_n(k)$. Here's a link: http://www.math.ethz.ch/~pink/ftp/LP5.pdf
In one sentence, the idea is to study these subgroups by looking at their "effective Zariski closures", whereupon techniques of algebraic geometry may be brought to bear on the problem.
A: A classic result in 3-manifold topology is Neuwirth's conjecture, which states that the fundamental group of a knot complement is a free product of two proper subgroups amalgamated along a free group. This was proven by Culler and Shalen using the algebraic geometry of representation varieties of 3-manifold groups into $SL_2 C$. Since this is an affine variety,
one may associate at least two ideal points (the non-triviality of the representation variety follows from Thurston's geometrization theorem). Associated to these ideal points is an action on a Bass-Serre tree, and then a technique of Stallings associates to this a separating (for at least one ideal point) surface with boundary, and the desired amalgamated product. 
A: Given a convex polytope whose facets are simplices, define the f-vector by f_i = the number of i-dim faces. Which vectors of integers are f-vectors? A list of conditions was conjectured, proven sufficients by direct construction of enough polytopes, and proven necessary by applying hard Lefschetz to the (rationally smooth) toric variety associated to the dual polytope. (A combinatorial proof came later.) See Fulton's book on toric varieties.
A: 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.
A: Of  course, the Cayley Hamilton Theorem is not really hard, and there are many many proofs of it. But you have to admit that, at least when you first step into linear algebra, it's rather surprising that it's enough to proof the theorem for diagonal matrices (which is a very short calculation). Because then you can derive it for diagonalizable matrices, which are dense with respect to the Zariski Topology (assuming w.l.o.g. that the ground field is algebraically closed). The latter is because every non-empty open subset is dense, a rather strange but here very useful property.
The same procedure applies to other polynomial identites in linear algebra, for example that the characteristic polynomials of $AB$ and $BA$ coincide.
A: Explicit theta function representations of soliton solutions to the completely integrable models (the work pioneered by Dubrovin, Matveev, Novikov, Krichever, McKean...).  
In the simplest case of the KdV equation, solitons are obtained from a singularization of the corresponding hyperelliptic curve. The kdV equation was originally derived to describe waves in shallow water and presumably had nothing to do with hyperelliptic curves.
Edit. And eventually the connection worked in the opposite direction as well: a solution of Shottky's problem which exploited the integrability of the Kadomtsev–Petviashvili equation.
