What are the most obscure or advanced mathematics with practical application

Question

Throughout my engineering studies there were jokes made by my professors (mostly mathematics professors) that referenced the fact that pure mathematicians strive to create mathematics with no practical application. Then a physicist or engineer comes along and finds a use for it.

I know that advancements have been developed for String Theory (maybe the only useful thing to come out of String Theory). But, in that vein, what are some of the most advanced or obscure mathematics that have real world, practical application to engineering, economics, computer science or such (especially if they are not well known)? And what branch of mathematics do they belong to?

Answer 1

I would actually go with the quaternions $\mathbb{H}$. They form a 4 dimensional, associative division algebra. With the basis $i, j , k, 1$ which satisfies $$ i^2 = j^2 = k^2 = ijk = -1 $$

They first might seem not useful at all, until you notice they are easily created with matrices, what means they are easily computable.

Quaternions are used to compute three dimensional rotations and thus are used in many (graphical) software frameworks.

Answer 2

Perhaps a bit obscure -- finite topological spaces applied to digital analysis

Perhaps a bit advanced -- spectral sequences applied to physics

Answer 3

A good candidate would be elliptic curve cryptography.

This is a direct practical application of finite fields, number theory, and other arithmetic geometry, that you would otherwise think have no purpose outside of pure mathematics.