During the past one and a half years I’ve spent as a graduate student, I’ve developed a few principles. I have decided to post them here, and update them as I learn more.
1. Don’t be afraid to branch out and learn subjects in other disciplines. Broadening your interests is easy in graduate school, and you’re paid to do it. Further, you may discover that you’ve gone to grad school for the wrong subject. You may be studying number theory, topology, etc., because you’re already good at it, and you don’t know enough about other subjects. It’s easy to get lost in the achievement factor in math, and just because you can solve problems with ease, doesn’t mean you’re truly motivated by the subject’s goals. The best way to discover your true motivations is to think about the guiding problems in the field and decide if they’re interesting to you in their most isolated form (e.g. do you “care” about distinguishing smooth structures on manifolds?). If the subjects no longer seem interesting, then move on to something you really enjoy.
2. Talk to everybody: professors, graduate students, undergrads, etc. The best way to learn is from other people, but don’t depend on them. Everybody has a tool they’re fond of — learn it. Find out what professors are interested in and ask for a research project or a reading course in that topic. Professors generally want to speak with you after you pass your quals. It’s part of their job to graduate students. So don’t be discouraged by professors that turn you down, five no’s and a yes is still a yes.
3. Don’t be afraid to say you don’t understand. Once you realize you don’t understand, you’re in the best position to learn. Further, many students may give the impression that they truly understand a subject, but will be forced to reevaluate if you ask enough questions.
4. Don’t be afraid of the computer. There is a massive amount of data in the world, and you’ll be at a supreme advantage if you know how to compute with it.
5. Go to seminars and colloquiums in your field and other nearby fields. Learning what other people are working on is likely to give you research ideas. Sign up to give talks in front of these groups, even if you don’t know a thing about the topic. Put a lot of effort into explaining concepts clearly and accurately.
6. Think long term. I know that many PhD students are only thinking about the next five years of graduate school. Assuming that everything will “work out” will put you at a severe disadvantage. Before every major choice you make in grad school, ask yourself why you want a PhD, and how will this choice benefit your goal. No matter what field you plan to go into, ask yourself the following question: who are the experts, and am I good enough to take their job? There isn’t a lot of room at the top, so you’re likely going to have to push someone out of the way to make room for yourself.
7. Schedule enough down time so that you can be fresh every day. Make sure you attend social functions, go to the gym, or play a sport, etc.
8. Learn to manage your time effectively. How many projects are you working on? If you get side tracked on a particular problem, move on and come back to it. Try to skim a book before you do the problems. Then, go back and read carefully when you need a deep understanding.
9. Learn how to measure success. Set benchmarks for yourself. Did you achieve more today than you do on average? Did you master a difficult concept? Try to achieve more every day, but don’t be discouraged if you can’t. You can’t learn everything in finite time. Some things are long term projects and even reading a few pages can give you something to be proud of.
10. Apply for fellowships. Teaching eats up a lot of time, but it does give you valuable experience. If you can’t get an RA, having your own funding can make your research a lot more productive. Having your own funding also increases the chances of a professor working with you.