Well, this was quick :-)
I don't have enough points to edit the wiki articles, so I guess I'll just toss a few here to get merged into the wiki articles:
math.NT: Cryptography (of course). A more quirky one is SETI (the primes in binary would be a very clear indication of a signal non-natural origin)origin, and would be a starting point for communication).
math.CA: Fourier analysis allows one to precisely divide up the electromagnetic spectrum, leading of course to radio, television, wireless, and so forth. MRI is based on inverting the Radon transform.
math.AP: Radar imaging is based on solving an inverse problem. The recent buzz about metamaterials and invisibility is based on understanding variable-coefficient elliptic problems.
math.CO: Group Combinatorial group testing allows one to quickly isolated defects in a large collection of samples by testing batches of samples at a time.
math.IT: Compression; efficient use of bandwidth
math.MG: Discrete sphere packing solutions lead to error-correcting codes.
math.PR: Here's a somewhat frivolous one (but one that casinos greatly care about): the number of times one needs to shuffle a deck before it truly randomises.
There's a way to use finite fields to multiplex cell phone signals so they don't interfere with each other, but the precise connection escapes me at present.
math.ST: Accurate polling; testing of medical treatments; fraud detection (e.g. via Benford's law)