Applying to grad school may seem like a long and complicated process, but I will attempt to present a simpler and clearer approach to the problem here.
It is evident that in order to achieve the best possible result from grad school applications, one must first rank the graduate programs in order of personal preference, and then do everything possible to be accepted into these listed graduate programs. In this guide I will be assuming that you already have your personal preferences sorted out, and that it only remains to increase your chances of acceptance.
Your one take-away from this guide should be that graduate programs look for these categories, in order of importance: (1) Recommendation letters, (2) GPA, (3) GRE test scores, (4) Personal statement, (5) Extras (to be described below).
These components will be addressed in order in the sections below.
A good recommendation letter is determined by a) the level of fame of the writing professor, b) how well the professor can attest to knowing you, in a research context as well as in the classroom, and c) how good the professor’s opinion of you is. Obvious steps that can be taken to improve these characteristics are: asking them to chat over a meal/tea if you are a senior; picking a famous professor and taking lots of their classes if you are a sophomore/junior. Most people end up getting 3 letter writers: their senior thesis advisor, an REU advisor, and another professor from classes.
In terms of logistics, it is good practice to ask your letter writers 3 times: once in person end of junior year (for politeness), once more by email near September-ish (to remind them, and check if their email works well), and lastly right before you send them all the links (to give them deadline information). It’s good to give your writers a full month to write their letters as some professors can be rather slow.
There’s not much to be said here, other than the fact that grad programs probably don’t look at your non-math GPA very heavily. You should also give as much information as possible on the application forms, if there is room to describe your courses — many Princeton 300+ level courses are common graduate courses elsewhere, so it’s worth mentioning the textbook you use.
In terms of logistics, most grad programs WILL accept the electronic version from SCORE → Student Center → View My Academic Record. For those that require a mail version, use SCORE → Student Center → Request Official Transcript.
GRE test scores
This doesn’t need to be a spectacularly fancy piece of writing. It should show that you can write in English with decent grammar and vocabulary. The format can be varied but I found a letter format was easiest.
You should cover a) what you thought about your math courses, while also explaining any holes or strengths in your transcript; b) your research experiences; c) your expected field of interest (naming a specific field is preferable to saying “undecided”); d) how this particular graduate program appeals to you. If you so desire, feel free to e) describe what you think your research strengths are; f) *sparsely* mention professor names at Princeton and at the target school; g) TeX it.
To explain d) it helps to ask around other schools for what the “feel” of their program is like. To give some brief examples, Stanford’s geometric analysis and algebraic NT; Princeton’s harmonic analysis, analytic NT, and probability; MIT’s combinatorics; NYU Courant’s PDE; Columbia’s geometric analysis.
In terms of logistics, once you write up your first one, you can resort to changing (d) and (f) while leaving the rest the same. Thus, a worst-case procrastination schedule is: ~2 weeks before your first deadline, write up a version and ask a friend to suggest some edits; correct and re-edit over 10 days; write up the (d) & (f) changes ~2 days before each deadline.
Deadlines start late November/early December.
“Extras” to your application include Putnam and IMO performance and published or submitted papers. These do add to your application, and can be listed in a brief resume (there’s usually a spot on the application form for one).
Hope this helps clear up some confusion, and good luck!
-Bowei Liu ’13