[miscPerson]

I only really know about the company-sci touching parts, but —

Mathematics didn’t have a strong notion of proof until the late 1800s/early 1900s. Contradictory “proofs” in calculus created a need for a more rigorous system of reasoning. (It took until the 1950s to formalize what an ‘infinitesimal’ is — but those were used in early calculus prior to limits.)

That gave us electronic computers, as the effort to reduce proofs to things that could be mechanized bore fruit. Every program is a proof!

Since then, the idea of proof and structuring mathematics has been radically redefined — introducing things like category theory and type theory as alternatives to set theory, while studying the impact of certain axioms and developing tools around “reverse mathematics”, which is fitting a set of axioms to the theorems you want to be true.

Modern mathematics is working on the study of logic topology, extending our reasoning tools to deal with complexities around proving equality in the hopes we can automatically verify mathematics. These tools overlap heavily with AI research.

The applications here are what you might guess: new AI methods, data analysis methods, verification of software, etc. The DOD has paid a fair bit to support that research into software verification, for instance. The NSF and others fund “big data” analysis.

I’m less familiar with other parts of mathematics, but —

The study of knot theory has impacts for physical sciences, from particle physics (anyons) to fluid dynamics. In particular, there’s some work to be done in higher order knot theory and computational knot theory (ie, how to be efficient). MSFT is building a quantum computer based on this — and has suggested that higher order knots might not require as expensive of hardware.

In number theory, we still don’t know much about almost all real numbers. Things like Chaitins constant exist — and such uncomputable, normal numbers form the bulk of the reals — but we don’t really know how to get our hands on them. In less exotic research, elliptic curves are used in cryptography. There’s some work I don’t quite understand in building out a homomorphic encryption — where we can operate on encrypted data.

In many areas, we’re still working through the algebra-geometry correspondence, which we got hints of 400 years ago but only formalized once topology and category theory were invented — and still are building tooling around.

And there’s lots of areas I have no idea about — but I assume are being similarly productive.

Ones that I know of, but can’t comment on: fractals, differential equations, bifurcation theory, and chaos theory.