Of course, this process could go on endlessly, but I think there was an important point that Noah didn’t emphasize enough: talk to people.
There are a few categories of talking that deserve special attention.
- You should make a point of going to conferences whenever possible (it can be extremely easy to get travel money for conferences as a grad student), even if they’re not exactly your field. If you have something to speak about, and can get a speaking spot, even better. If you’re wondering how one goes to conferences, there’s a simple algorithm.
- read the AMS math calendar
- request funding for any ones that sound interesting
- rinse and repeat.
- You should do whatever you can, non-annoyingly, to cultivate relationships with mathematicians, especially ones who are older. They can give you valuable advice, serve as good references, and can be good collaborators.
I feel like it can’t be emphasized enough: mathematics is a social activity. You’ll never learn it properly from books and papers, and you can’t rely on your advisor to tell you all the things you need to know. Rather, you have to talk to the people around you, and make sure you have people around you to talk to.
Of course, different levels of talking are good for different people. I’m a pretty sociable guy, and that shows in my mathematical work (it’s been almost 3 years since I’ve written a solo paper and don’t have any on the horizon), but even if you don’t want to collaborate with people, you really do need to talk to them about math.