Yakk wrote:It is, in a sense, more of a sign that esoteric pure mathematics ends up having applications despite the best efforts of the subject area than evidence that it is applied mathematics.

This.

For example, one of the areas that I'm rather interested in and have at least a vague concept of an idea about what's going on is set theory, model theory and logic. One would think that such a highly abstract and in some instances bordering on philosophics area would be as pure as you can get. But - it's simply not true these days if you consider that some of the results there have more or less direct effects on areas in, let's say, complexity theory in CS. And that, in turn, can have practical consequences for very applied algorithmic questions. Or consider proof theory and its relation to validation of programs and the implications on what's possible there or not.

Another example just came up in a

different discussion on this forum that kinda stunned me: graph theory. Now I know that graph theory, by essence, is a very applied topic. But when doing some research on directed graphs, it turns out that most recent and current results are about finite directed graphs, which often are very directly applied questions.

And then there's what's these days is often used in jokes: Whatever "pure" math comes up with is getting turned into something "applied" by theoretical physicists. It's hard to find a field that does not have some "applications" over in theoretical physics.

A reason that I believe is relevant is something that emerged over (and that's a vague estimate, correct me if you know better!) the last half century or so, which is a focus in pretty much all math towards generalisation and dualities between (a priori seeminly unrelated) fields of mathematics. Whether you look at the infamour proof for Fermat's last theorem (connecting seemingly very basic number theory in a fundamental way to something as abstract as modular forms) or Diophantine equations paired, through algebraic geometry, with cs (

Matiyasevich) or consider what economists do, which includes functional analysis and various differential equations.

These days, it's challenging to find an area of mathematical research that doesn't have any known relation to something more "applied" than what "pure mathematics" would have meant 50 years ago. I'm not aware of any, and I'd be curious if anyone could name one.