paper, like.. pieces of paper.. yes, that is old-fashioned! however, in deep math, maybe this is something.. I was told after attending a world-class lecture week in pure math, that it is common for PhDs and post-docs in math to have very little to talk about between each other.. since their interest and subject area is almost certainly deeply buried in some specialty. Aside from the quip "all roads lead to cryptography" perhaps just an excuse for certain people, I can see how this "different worlds" problem in post-doc math might be real..
The fellow writing this post is first-year however.. maybe its different.. Also I just cloned and built pdfgrep .. I am definitely going to try that out immediately .. thx!
I keep questions, slides, written notes, etc. controlled and semester-specific things in a directory that isn't tracked.
To me, the place for the real time savings is organizing your teaching workflow. If you're a grad student, make sure to get on top of that ASAP. A lot of it is very repetitive both within a given semester and also from year to year. I have a ton of scripts built up over the years for things like organizing lecture notes (especially if you are teaching a class you have taught before, you don't want to have to rewrite all your lectures), creating problem sets / solutions, inputting grades into the LMS,... I even have automatic solution generators for a few rote calculations that I assign many problems about (Gaussian elimination, for example).