As a PhD student in applied mathematics or mathematics in general, are you expected to be able to prove every problem, for example, in an elementary real analysis book? I know it sounds silly but I am wondering if I have high expectations of myself…
For example, some theorems in an elementary real analysis book have proofs which are lengthy and time consuming to understand. Still I try to understand them and manage to do so. But if you ask me about the theorem, say after 2 months, there is a high chance I’d have forgotten how to prove it. I might have ideas but I cannot solve it in totality.
What I’m asking is, I guess, is it normal to understand something in math and forget about it? Or does the fact that you forgot about it/how to do it an indication of not having understood the subject in full in the first place?
Absolutely! When I look back at some of my old papers, I find it hard to follow what’s going on.
The point about research is that you’re pushing yourself and mathematics to its limits. It’s impossible to maintain such a high standard over such a bredth of knowledge.
PhD research requires independent and original thought. No-one cares if you can solve all of the problems in someone else’s book. What good is it to you anyway?
Insight, vision, imagination, understanding and passion are what are needed to be a successful researcher.
It’s not about being a memory man who can memorise the 1,001 tricks required to solve various problems. As long as you understand the material when it’s infront of you, you’ll be fine. You’re not expected to remember millions of pages of trivia; that’s what libraries and journals are for!