I must have not communicated what I meant clearly, because you missed my point.

Only a very, very small fraction of results in number theory are applicable. Only a very, very small fraction of stuff done in algebraic topology is applicable. While descriptive set theory is indeed sometimes applied in functional analysis, the intersection between these two is not a significant fraction of each one of them, and by the time you apply (a very small fraction of) functional analysis to statistical physics, you're already too far from descriptive set theory to even see it on the horizon.

I'm not saying that none of the mathematics is applicable, because this is obviously not the case. What I'm saying is that the stuff that gets applied to real life problems is surprisingly small, even more so when you're not a mathematician.

When I first started to learn mathematics, I was completely overwhelmed, when I found out just how much knowledge is out there in this field of human activity. The planes of mathematics are so vast, there's almost nothing else in sight when you stand atop of the Mount Bourbaki. I realized that even if I get a PhD in pure mathematics, I will still only be able to learn less 1% of mathematics ever created in my whole lifetime. That's why when I hear people saying that all math can be applied, I think that they must have not realized just how much of the stuff is in there.

I used to specialize in algebraic topology, and when I first learned about persistent homology, I encountered a home page of the professor at some US faculty, who specialized in applying algebraic topology in real life, and the first reaction of me and my classmates whom I have shown his website was not appreciation of his results. We were totally amazed that this stuff can be applied to anything _at all_. Yeah, some of these applications were to the problems that are usually solved in a better way, some problems were very contrived and seemed to actually be tailored so that one can apply algebraic topology to them, but still these were very fine and interesting results, and we were very surprised by them.

The field of algebraic geometry actually makes an interesting example. Indeed, it is considered by many one of the most, if not the most abstract field of the mathematics. Initially, at the beginning of the previous century, people were mostly concerned with studying the sets of solutions of systems of polynomial equations. David Hilbert with his landmark results being his basis theorem and Nullstellensatz laid fundamentals to this field, and because of an essential assumption in the Nullstellensatz theorem that creates a bridge between algebra and geometry, for many decades most of the results concerned only polynomials and sets of solutions in algebraically closed fields, because for anything else, the apparatus was just lacking. Algebraic geometry didn't have a reputation of an extremely abstract field, especially since sets of solutions to systems of polynomial equations are quite natural objects, and it's easy enough to imagine their occurrence in real life problems.

In 1950s and 1960s, though, the field was completely and utterly revolutionized by Alexander Grothendieck and his school, and that's when the field gained its fame of being very esoteric. Grothendieck's methods and approach allowed algebraic geometers to tackle vastly bigger range of problems, and ultimately to efface the distinction between algebra and geometry, at the price of making things much more abstract and distanced from more concrete considerations. That's when algebraic geometry expanded its reach, to encompass a large amount of research in abstract algebra and number theory.

In the meantime, another interesting thing happened in the field: the advent of computational techniques. Things like Groebner bases really pushed things forward and made a lot of theoretical stuff actually doable in practice. This is mainly what made things you mention in your post possible: while many industry problems could be formulated in geometric terms earlier, only rise of computational methods actually allowed us to solve them.

The point here is this: algebraic geometry field consists of two parts, the older and more concrete, which can and is (relatively) frequently applied, and the newer, but more abstract, of which applied is very little -- with notable exceptions though being for instance finite elliptic curves with well known application to cryptography, or Calabi-Yau manifolds which are intimately connected to the string theory. Nevertheless the applied stuff constitutes only a small part of the field, the rest is just pure and abstract mathematics, without any applications whatsoever, though it's still worth noting that algebraic geometry is still relatively very good in the amount of applicable stuff, and fields like algebraic topology, not to even mention descriptive set theory, fare much, much worse with regard to it. For me, a bit naive is to think that big part of mathematics will be applied, when there's so much of it.