I know, I know, there are lots of advice pages out there, for example (my favorites) this extremely broad, kind and generous page by Terry Tao or this very focused one by Doron Zeilberger. But like any advice giver who respects himself I am convinced that I would do you, young mathematician, a great wrong if I don’t share what I have learned in my journey so far, and that this is an urgent task. After all, I thought about this yesterday.
So here is my first piece of advice: Stop doing things.
Let me explain. There are some things which you have to, at some point, stop doing, or else you will never have time to make progress.
- Stop reading all the preliminaries. Of course, you want to have all the background before you start your next research project. I had a friend who spent the whole two years of his master’s degree reading. There is always another book that really explains something mentioned in the current book that you are reading. There’s an infinite chain of prequels, and no end to it.
- Stop trying to understand every single word in every paper. Some papers will not deliver what you want them to, or will not be interesting, and you should try to figure out if this is the case before you crack your head open on Lemma 4.
- Stop going to every talk in your department. It’s OK, they were a little surprised to see you there in the first place.
- Stop spending two hours to prepare every one hour lecture. It’s unsustainable.
- Stop working on the problems from your thesis. It was very good to spend a few years working on the same problem, because that way you were able to obtain some really great results. But even though there are some tiny corners left to clear, it is time to move on to another problem, open your mind, maybe even change your field.
- Stop changing your field. It really takes some time to get a grasp of a research area, it takes time to make some real impact in some field, it takes time to get to know everybody and to get everybody to know you. It is a shame to throw this away too fast because another problem looks sexy.
- Stop trying hard to keep up with what is going on in your field on a daily basis, (like reading every paper that comes on the arXiv). That’s what conferences are for.
- Stop asking everyone about your research questions. They never know the answer. It is really for you to discover.
- Stop trying to collaborate with your grad school buddies. Though many collaborators become good friends, not all good friends become good collaborators.
- Stop reading advice that other mathematicians give. At some point you have to start writing these.
Now, I hope you understand what I mean. It is crucial (listen carefully graduate students, it is crucial) that you actually stop doing these things (thus you have to start doing them at some earlier point), if you simply never do these things to begin with then I think that you are on the wrong track.
Perhaps after some time goes by you have to start doing these things again. That does not contradict the fact that sometimes you just have to stop.