> Defining "applied math" as mathematics that has found applications is disingenuous, IMO; what is and isn't applied can change rapidly in the course of a few years.
Not "disingenuous" at all: I want to apply math, especially to making money. To make money is the main reason I studied math; I wanted math to help my career, to make money, to support a family. If not math, then maybe physics, some part of engineering, etc.
> what is and isn't applied can change rapidly in the course of a few years.
My experience is that that doesn't happen very often. But again, once again, over again, yet again, I am not, Not, NOT, N.O.T. -- clear enough -- running down pure math. NOT doing that. I just wrote that foundations as in the OP was too far from applications for me. I have been totally, overwhelmingly, crystal clear about this point in this thread, and there was nothing in my first post that ran down pure math. Again, yet again, ..., I was talking about my view for me of foundations such as in the OP. You misread my first post and didn't read my other posts.
If someone found some applications for PCPs, then good for them, and that makes PCPs applied math.
Better techniques for proofs of correctness obviously would be good work in computer science with plenty of valuable applications.
> What makes an industry with a lot of money in it, like online advertising, worth more than study of something completely theoretical?
Nothing, but most sources of financial investment want a financial return. That's much of how our economy works.
To be blunt and frank, Congress votes money for math mostly for US national security and because of the role of math in WWII and off and on since then. If some people at the NIH tells Congress that they like math, too, then that will help. But the remark at the beginning of the movie on Nash is basically correct: "Mathematics won WWII". Congress can be slow on the uptake, but they tend to believe that math did win WWII and had a big role in GPS, Keyhole, stealth, design of the core of fission and fusion bombs, etc. Congress takes US national security quite seriously. The other biggie for Congress is the NIH -- if only because most people in Congress have gray hair and want research that will help them if they get sick.
It remains, if you want a job in math, especially connected with computing, then those jobs are still where I started my career (and should have stayed there) within 100 miles of the Washington Monument.