big list - What are some applications of other fields to mathematics?

I can think of at least three things that the question might mean, and it would probably help if Steve clarified which ones count for him!

(1) Other fields suggesting new questions for mathematicians to think about, or new conjectures for them to prove. Examples of that sort are ubiquitous, and account for a significant fraction of all of mathematics! (Archimedes, Newton, and Gauss all looked to physics for inspiration; many of the 20th-century greats looked to biology, economics, computer science, etc. Even for those mathematicians who take pride in taking as little inspiration as possible from the physical world, it's arguable how well they succeed at it.)

(2) Other fields helping the process of mathematical research. Computers are an obvious example, but I gather that this sort of application isn't what Steve has in mind.

(3) Other fields leading to new or better proofs, for theorems that mathematicians care about even independently of the other fields. This seems to me like the most interesting interpretation. But it raises an obvious question: if a field is leading to new proofs of important theorems, why shouldn't we call that field mathematics? One way out of this definitional morass is the following: normally, one thinks of mathematics as arranged in a tree, with logic and set theory at the root, "applied" fields like information theory or mathematical physics at the leaves, and everything else (algebra, analysis, geometry, topology) as trunks or branches. Definitions and results from the lower levels get used at the higher levels, but not vice versa. From this perspective, what the question is really asking for is examples of "unexpected inversions," where ideas from higher in the tree (and specifically, from the "applied" leaves) are used to prove theorems lower in the tree.

Such inversions certainly exist, and lots of people probably have favorite examples of them --- so it does seem like great fodder for a "big list" question. At the risk of violating Steve's "no theoretical computer science" rule, here are some of my personal favorites:

(i) Grover's quantum search algorithm immediately implies that Markov's inequality, that

$max_{x in [-1,1]} |p'(x)| leq d^2 max_{x in [-1,1]} |p(x)|$

for all degree-d real polynomials p, is tight.

(ii) Kolmogorov complexity is often useful for proving statements that have nothing to do with Turing machines or computability.

(iii) The quantum-mechanical rules for identical bosons immediately imply that |Per(U)|≤1 for every unitary matrix U.