"Please follow this very smart Russian proverb: 'Measure 7 times, cut 1 time'. It has the English analogue too: 'Look before you leap'. These words of wisdom are of great use in applied mathematics since we are the people who actually go out, look at the processes in nature, try to extract the essential features out of them and summarize them in precise mathematical terms. Usually, there are myriads of ways to write down the models that describe one and the same reality but we have to focus on only one of them. The point is to focus on a good one (and not the one that pops in your mind first!) before starting to put a lot of effort into the mathematical analysis. Study the literature, talk to people and, most importantly, think (as hard as it may be... ;) BEFORE you waste your time and effort."

~ Sergey Kryazhimskiy

1) Enjoy the curricular freedom PACM provides and take the courses and work on the research projects you choose for yourself.
2) Start working on research projects early, in the first year, either with small projects with professors you are considering for advisors, or directly on your thesis research if you know what you wanna do. Think of having an extra (demanding) class *every* semester and do not get bogged down in class work.
3) Go to seminars/classes in fields other than yours.
4) Find a few professors to form closer collaborations with, other than your advisor, for example, your prelim examinors. Take their classes and discuss your research interests with them.
5) Make sure you put a little of yourself in your research work: Do not let your advisor make you a puppet doing the tedious calculations he/she does not want to be bothered with!

~ Aleksandar Donev