Question

Which areas have the most 'adequate' level of abstraction in Modern Math?

Answer 1

For what you're describing, I think differential geometry fits the bill very well. Can be pretty abstract and hard to get into, may be very technical, but I doubt that most people would think of it as the height of abstraction in maths.

Vast applicability in physics - general relativity, gauge theory, classical mechanics and continuum mechanics usually forwarded as the crown jewels. Lie theory and representation theory. Geometric analysis, in which we've seen some of the most impressive PDE methods with stunning results in the past few decades, especially relating to topology. Complex geometry, where there is particularly rich interaction with algebraic geometry.

Dynamical systems. Optimization on manifolds. Discrete differential geometry, which is of interest in computer graphics and design, as well as in combinatorial surfaces I suppose.

I've seen differential geometry getting talked about in the context of machine learning (manifold learning?). There's apparently stochastic differential geometry as well (think Brownian motion on Riemannian manifolds).

Answer 2

First, as a side note, differentiation in TVS is used all over the place in differential geometry. This is because it's very natural to treat certain spaces of functions, vector fields and the like as infinite dimensional manifolds and many functionals can be seen very naturally using those.

Anyway, as other have mentioned I think diff geo is definetely a subject to look at, it's a field that definetely isn't shy to seek and use all of the fancy abstract nonsense gadgets, but is still very concrete in many cases, is filled with a lot of (hard!) analysid and pdes, and you can actually draw pictures which is kinda nice.

Answer 3

Have you read Williams - Diffusions Markov Processes and Martingales? Because I have never read a book where the underlying material is so highly applied (probability) yet the book was so theoretical and so extremely generalized that the material felt as abstract as one can possibly get.

Answer 4

Maybe I’m misunderstanding the question but combinatorics? In particular, extremal combinatorics. A lot of interesting questions in extremal combinatorics involve generalising known results as far as possible. But it’s still fairly concrete in its relation to computer science.