Please share any general tips or advice you have on doing mathematical research.
How do you identify good problems to work on or to think about? What do you do when you get stuck on a problem? Etc.
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My 2c worth of personal perspective:
On problem selection - let the problems select themselves. In other words, when you have nothing to think about, talk to as many people as you can and read as much as you can in and out of your area of interest. Sooner or later a problem will emerge that doesn't let you go. It may not be an explicitly conscious choice but it will happen, and you'll just find yourself trying to solve it without ever having decided to do so. Realize that, and work on that problem.
On being stuck: One of the things that helped me most consistently is (my version of) Polya's dictum: inside every problem you can't solve lies a smaller problem you also can't solve. Find it!
One thing I find incredibly useful is the MathSciNet reviews. Often the reviews make the paper more clear. If the paper doesn't address the big picture, the review will often give context or, if the result fits into a larger program of research it will mention this. Sometimes the review highlights unique technical arguments.
Also, everything is hyperlinked, in that you can pull up all articles referenced by a paper or, better, all articles (or reviews) that reference a particular paper (this has often been useful when I didn't understand something in a paper or didn't know the point of a paper). I spent a lot of time reading reviews as a beginning grad student and it helped me get a sense of what questions people find interesting.
Some of the best advice comes from Richard Hamming's talk You and Your Research.
Terry Tao has some helpful ideas.
I can't say I've really mastered the practice of choosing problems. I guess if there's an area of math that you're really familiar with, questions will just start coming to you, but I can't tell how much of that ability comes from some osmotic process from the examples of peers.
When I'm stuck, I try to write down where I am in a lot of detail, then I take a break. If I'm lucky, I know what sort of knowledge might be able to crack my problem, and if I'm even luckier, I know who has that knowledge.
Often, I find that the good problems come from errors in my work -- I write a proof of something that (at the time) I think is trivial and later find a counter-example. Suddenly, the problem is non-trivial and interesting.
When I get stuck on a problem? I work on another problem, a related problem, a broader problem, a more specific problem, etc. Anything to try to understand why it is difficult to solve this problem.
Some further advice and links can be found here.