>>11814592Geometric group theory
Langlands if you're allowed to take the time to teach him 2000 pages of prerequisite bullshit first
Related, gigantic chunks of representation theory are pretty new, e.g. geometric representation theory, most of the theory of infinite-dimensional Lie representations, D-module stuff, the entire idea of a quantum group...
Moonshine/VOAs. Related, you should probably tell them we think we know all the finite simple groups now. That was a big deal.
90% of the entire field of algebraic combinatorics
The foundationstards trying to do weird type theory shit
Definitely the biggest one I can think of, computing was still very primitive in 1970. There essentially wasn't even an Internet. Most of theoretical computer science didn't exist (I checked out of curiosity, P v. NP wasn't even formally stated until '71), and the things other not-TCS people work on today would have been literally just sci-fi in 1970.
And I'm rattling off stuff I happen to know exists. You unfortunately seem like you're more interested in analytic stuff, which I am as qualified as a poodle to start namedropping on, but I'm sure it's exactly the same over there.
The reason everything you know is old is because a Bachelor's exclusively covers old math. You can't do fancy new stuff until you've already learned the foundational old stuff it's built on.
