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What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, e.g. math.AT, math.QA, math.CO, etc.

This is a community-wiki question, so please edit and improve pre-existing answers: let's keep it to a single answer for each subject area.

(This is inspired by Terry Tao's recent post about a periodic table of the elements listing commercial applications. He suggested it might be fun to have such a summary for either the MSC top-level subjects or the arxiv subjects.)

I'd like to propose that for areas in which the applications are either numerous, non-obvious, or generally worthy of discussion, someone volunteers to open up a new question specifically about that subject area, and takes care of providing a summary here of the best answers produced there.