How do you retain years of graduate mathematics without constantly rereading entire books?

I'm a mathematics student, and after almost three years I've finally realized something that somehow nobody explicitly told me. (now starting masters)

When I started learning mathematics seriously, my attitude was basically:

Needless to say, that isn't how it works.

For example, I can read a beautiful chapter in Feller on random walks, understand every proof and appreciate all the clever ideas. Six months later, I still remember the big picture—recurrence vs. transience, gambler's ruin, reflection principle—but I've forgotten many of the subtle caveats, the exact hypotheses, and the elegant tricks that made the proofs work.

The same thing happens with algebra, analysis, topology, etc.

I've now realized that if I want to eventually do research, I can't afford to keep relearning entire books every year.

So I'm thinking of making very short recap notes after finishing every chapter, something like 2–4 pages at most.

My current idea is:

  • Definitions
  • Just list the term if I know it well.
  • Write the full definition if it's subtle or easy to forget.
  • Main theorems
  • Either just the theorem name (if familiar), or a concise statement.
  • Proof idea
  • One or two lines explaining the key insight or construction.
  • Not a full proof.
  • Key examples
  • The canonical examples from the chapter.
  • Connections
  • Where this chapter fits into the bigger picture.
  • e.g. "Conditional expectation → Martingales → Brownian motion."
  • Common caveats / mistakes
  • Hidden assumptions.
  • Places where people often misuse a theorem.
  • Technical hypotheses that are easy to forget.
  • Personal insights
  • Intuitions or analogies that made something click for me.
  • Exercises worth remembering
  • Only the ones that taught an important technique or idea.

The goal isn't to replace the textbook, but to create an "index into my memory" so that six months later I can review an entire book in an hour or two instead of rereading hundreds of pages.

For those of you further along (PhD students, postdocs, faculty), is this roughly how you maintain long-term mathematical knowledge?

Or is there a better system you've found over the years?

PS: I had used AI to polish my question.

Author: shashypants