If I was applying for a PhD today I think the topic I'd pick would be related to random matrices. The content of this paper illustrates why in the most readable summary I've come across:
https://www.imaginary.org/sites/default/files/snapshots/snapshots-2018-002.pdf
It (bizarrely!) connects the subject I did study (Riemann surfaces) with physics and random matrices and to keep the CS part of my brain happy there's a recursion scheme. It also hints at why I think String Theory is still an important subject despite me never having "believed" in it.
