Side note: If you're applying to grad school in CS, you're welcome to contact me, Tum ('14), at sorathan (at) cs (dot) stanford (dot) edu for more information. A useful CS PhD application guide from CMU that I usually point to can be found here. Note that if you're interested in math, there are also many opportunities to do that in CS as well.

GRE General Test

It would be smart to take the general test as early as possible, e.g. in the summer after junior year, the summer after sophomore year, even the summer after freshman year.  You'll be glad to have it out of the way in senior year.

The quantitative section on the GRE general test is nothing to worry about.  However, it would be a good idea to take a practice test to familiarize yourself with the kinds of problems and make sure you're not lacking in any areas.  Take the test slowly and methodically--the most difficult aspect of the test is refraining from constantly second guessing yourself; the problems are that easy.

On the other hand, the verbal section is hard.  You should study the vocab vigorously.  Remember that the general GRE is targeted towards the "average profile college student."  The vast majority of college students only took 1-3 math courses in their entire curriculum, while the rest of the time was spent doing reading and writing.  HMC students sometimes have less experience in that regard.

GRE Mathematics Subject Test

Take the subject test early and often--as soon as possible!  That extra chance could make the difference!  You can--and are encouraged to--take the test before your senior year, but it would be best to wait until after algebra I, analysis I, and discrete.  You also need to sign up ASAP.  Since there are limited offerings, they fill up really quickly at all reasonable locations.  This year there are tests on October 9, November 13, and April 9.  Mudd is hosting a test center at the November date.  Note however that many grad schools have GRE score deadlines on or around December 15, and the approximate score report mailing date for the November 13 test is December 24.  The registration deadline for the October 9 test is September 3, but registration is still possible until September 10 with a $25 fee.  For more information on the test deadlines, look here.

The best way to study for the test is to study definitions and drill problems.  The algebra and analysis questions tend to be of the "prove you really understand the basic definitions" variety rather than anything complicated.  For these, review the basic definitions.  These problems should be the fast and easy ones.  For the other problems: Practice Practice Practice!  The most challenging part of the test is the time pressure, so you should spend a lot of time working on solving problems quickly.  The part I was most annoyed by is the large chunk of AP Calc, basic multi V and basic DEs problems.  These aren't hard until you realize that you have less than two minutes to work out the computations.  My guess is that the best way to prep for that part is to just drill them--that and practice estimating so that you don't waste time on tedious arithmetic (you don't get a calculator).  In some cases, the most efficient way to approach the problem is to just check the possible answers and see which works.  They also like the "here is a nasty trig expression, which of the following is equivalent" problems.  For those you plug in pi, pi/2, pi/3, pi/4 until you narrow it down to one.  Sometimes just the first two are sufficient.   There are also a handful of random problems that they pull from assorted higher areas of mathematics.  There will probably be a complex analysis problem of some sort.  I think the test I took had a "here is an algorithm and an input, which of the following is the right output" type and something from set theory.  If you go poke around on the website you can probably find a list of the kinds of problems they can ask.  The biggest thing though really is making sure you can do the computation problems fast enough, or you are going to find yourself running out of time.

There's a good description of the specific topics on the GRE website.  Roughly, the topics are as follows:

1. Advanced math: basic group theory (groups, subgroups, order), analysis (think convergence, cauchy equation of real numbers, open sets, compact sets), combinatorics and number theory (modular arithmetic, bijective arguments, choose and permutations).  Many people won't need to study this stuff much at all.
2. Calculus: multi V, flux, div grad curl, green, stokes, interchanging integrals (when can you?) actually computing (indefinite) integrals, trig forms and identities, integrals of trig forms.  This will probably be the brunt of your studying for the math GRE because you have likely forgotten all your integral calculus, and especially your multi V.  This is where a test prep book comes in handy, it will contain tables of integrals for you to study and lots of practice problems.

As for drilling problems--which you really, really need to do--find as many old tests as you can.  There is at least one freely available on the ETS website.  Take them real time after reading through a prep book.  I really like Princeton Review's book.  I usually like them best.  I still use it on occasion if I need to brush up on undergrad math!  Another Mudder used both the Princeton Review book and the REA book, and found the latter much more useful for test prep.

Choosing schools to apply to

A new set of rankings of graduate schools has just been released.  It's rather complicated.  Here is the FAQ at the Chronicle of Higher Education:
http://chronicle.com/article/New-Doctoral-Program/124634/

 And here is a tool to explore the rankings:
http://chronicle.com/page/2010-Rankings-Doctoral/321/

Much of the data measured has more to do with faculty reputation and publication rate than with grad student experience.  But you should look carefully at the rankings for completion percentage, time to degree,and percentage of Ph.D.s with academic jobs.

Grad School Applications

From the perspective of someone who has recently applied to pure math graduate programs, the following are the main components to a typical application.  (Things may be different for students intending to enter applied math programs.)

1. Personal Statements/Curriculum Vitae (CV)
2. Recommendations (by far the most important component of the application--you need three or more)
   
3. Math Subject GRE, General GRE
4. Transcripts
5. Application Fees

Discussion:

1. Typeset everything in LaTeX if possible.  Some schools require you to have specific things in your headers/footers.  LaTeX will handle this nicely, plus the more TeX skills you show off the better.  Make a really nice CV.  This is different from a resume in that it may contain multiple pages, but as an undergrad, do not exceed 2 pages.  Look online for templates, ask Judy (Office of Career Services) for feedback.  Nearly every institution to which I applied allowed/requested submission of a CV.  An example of a nice CV is attached (see the bottom of this page).
   
   Usually you are required to submit a Personal Statement, Research Experiences Statement (and for the NSF a Research Proposal).  Sometimes a Diversity Statement is allowed, but not required.  You should have 1000 word (i.e. two single spaced pages) versions for most fellowships and grad schools.  Some may require 500 word versions.  After crafting these statements, ask your recommending professors to give feedback.  Go to the writing center.  Get lots of opinions!  Your grad school statements should sound a lot more professional and less touchy-feely than the ones you wrote for undergrad.  Math grad schools (pure math at least) will NOT care about your leadership skills (unless it is math related), religious affiliations, piano skills etc.  They don't want to know who you are as a person, they want to know who you are as a mathematician.  Everything in your statement should be math related!  E.g. what classes/theorems/problems inspired you, times when you got really excited about a problem and came up with a really elegant solution, your REU's, your work as a grader or AE math tutor, some details about any publications you may have etc.  You basically need to convey in a persuasive, professional, and not overly emotional manner that you will SHRIVEL UP AND DIE if you are not doing mathematics 24/7, what type of mathematics you could see yourself doing (with supporting coursework/experiences/etc.) and why you will thrive in department X.  Write them specifically for your top choice, modify them for the rest of the applications.
   
   The Research Proposal for the NSF application I think is weird.  I guess it would make more sense if I were a biologist and wanted to work in Lab X and was already doing publishable research in tissue cultures or something.  For math it is weird because almost no undergraduates are yet ready for graduate level research... or can even talk about interesting open problems in a way that does not completely belie their ignorance.  My only advice is to be completely honest.  Talk about a subject/problem that you have enjoyed working on, express an interest in developing your knowledge of that area, maybe talk about how aspects of it are accessible to undergrads and would bring together mathematicians at different levels (the NSF loves this sort of thing).  I do not recommend that you put a lot of specific "math" in it.  For example, if you did representation theory with Professor Orrison, don't put in 6 definitions and that huge 20x20 matrix you are so proud of.  Talk about the research in plain English, results, big picture things, applications to other areas of mathematics, who cares about it etc.  You want to give the NSF an idea of a problem you could see yourself working on, without being boring, too technical, or ignorant and obnoxious.  And while you definitely want to write about something you understand well enough to talk about intelligently, it is also good to be aware that this proposal is not fixing what you are going to work on for the rest of your life (or even the next several years).  From what I have gathered, they are more interested in seeing that you have the mathematical maturity to put together a research proposal than in whether or not that is the particular research you end up doing.  This may not seem like a big distinction, but keeping it in mind can help make writing the proposal seem less intimidating.  Apart from that, talk with your professors about this one.  (Professor Gu might be a good resource.  She worked for the NSF.  I doubt she read grad school proposals, but still she must have a wealth of information.) This is an excellent presentation on how to apply.
   
   
2. Your letters of recommendation are far more important than your GRE scores.  You need to think very carefully about who will be writing your letters.  In fact, when you ask people to write your letters, you should ask them the following question:  "Can you write me a strong letter of recommendation?"  This won't offend your letter writers, and they will tell you if they can't.  But make sure you're getting good letters.  If you have a mediocre GRE score and amazing letters, you have a much better shot of getting into a top-tier school than if you ace the GRE and your letters suck.  I can't say it strongly enough:  If you do nothing else on your application, get good letters of rec.
   
   You will need at LEAST 3 letters of recommendation.  Some schools allow you to submit more which may or may not be reviewed.  If you can get more than three (say x_1, x_2, x_3, x_4), then you will have a choice.  For example, if professor x_3 can give you an amazing recommendation, and professor x_4 can give you a decent recommendation but got his/her Ph.D. from Carnegie Melon, then in your application to that school you may want to use x_1, x_3, and x_4's letters.  Yes?  I only had 3 letters myself, but you can see the possibilities.  It is really important to begin writing the items in (1) ASAP!  When requesting letters of recommendation, use Professor Orrison's guide.  Also, you will want to send them drafts of the items in (1) and politely ask for feedback if they have the time.  Select recommenders who know you, in whose courses you have performed well, and may have contacts at the institutions in which you are interested.  Try to have at least one non-Mudd recommendation.  This is where REU's come in handy.  Your adviser from an REU should be happy to write you a letter.
   
   
3. See above on how to study for GRE's.
   
   
4. See the checklist below.
   
   
5. Remember the GRE's are expensive, and all these applications have fees ranging from $50-$100.  I budgeted $2000 for this lousy application process (I applied to 12 schools).  Start saving and/or begging!

To give you an idea of how this all fits together, here is a grad school checklist.  These are just my opinions.  Obviously your advisor's opinions trump mine.

1. Focus.  And then do this stuff.
2. Go to this link.  Pick at least three from the top 20 that you weren't planning to apply to and add them to your list.
3. Once you have the list of schools you think you are going to apply to, go talk to your advisor and/or any other math prof that knows you well.  They can help you make sure you are applying to a reasonable number/set of places and might well point out some awesome program you have overlooked.  They can also likely tell you which profs have connections where.  This can be useful information when deciding who you want to have write recommendation letters for you.
4. There are some schools where the deadlines for you getting in your part of the application, your recommendation letters arriving, and GRE scores arriving are not the same - make sure you understand exactly how the deadlines work at each school.  In a similar vein, it is not always the case that all of your application materials are going to be sent to the same place.  It is quite possible that some things will be sent to the overall graduate school at the university in question and that other things will be sent directly to the math department.  This boils down to being careful about reading directions, but it still is important to be aware of and occasionally you do have to hunt for the directions.  There are some schools where I started to wonder whether proving you could successfully navigate their mess of a process was actually part of the application process itself.
5. Write down, in a spreadsheet or word document, NSF fellowship, NDSEG fellowship, other fellowships to which you are applying, your 10+ schools and all their due dates.  Pick out the ones with due dates in mid-December and do the following ASAP:
   
   1. Register to apply.  (I.e. get username/pw on their site, begin application.)
   2. Register your recommenders and make sure the application sends emails to all of them.
   3. Find the address to which you send supplementary materials like transcripts.  Write all the addresses down.  When you have all of them, send the list of full addresses to transcriptsetc@hmc.edu along with a request to send TWO copies of your transcript to each school.  (You need to send the email from your HMC email account, and you should include your name and class year.  See here for more information.)
      
   4. You can only send GRE scores automatically to (I think) 3 schools.  For the rest of the schools, go to the ETS website and attempt to sign in to send your GRE score reports.  If it won't let you, call the number that it tells you to call and ask them if you can get your scores AND (even if you can't get your scores) if you can have them sent to more schools as soon as they're ready.  Specify your list of schools and have them sent to the math departments.
6. Go read the Statement of Purpose prompt at one of your top-choice schools and begin writing.  You can use essentially the same statement for all of your schools, but it needs to be good.  It should be two pages single-spaced for most schools.  This is one of the most important parts of your application.
7. Whenever you have a free moment, go to one of the schools to which you are applying and fill out information.  It's really pretty straightforward and simple for the most part, but it's gotta get done.

Questions to discuss with your adviser when you are thinking about grad schools

 

1.     Is the geographical location of the school important?

2.     Is prestige important?

3.     Do you want a department that is primarily pure, applied, operations research or statistics or do you want a department with faculty in more than one of these areas?

4.     Is your GRE score strong?

5.     Do you want to work with certain faculty in particular?

6.     Do you want to choose an area of focus before you begin or remain open to multiple options?

7.     Do you care about the size of the school?

 

 

Questions for when you are looking at a particular school.  Ask current grad students as well as the Director of Graduate Studies (usually a faculty member):

 

1.     What are the coursework expectations?

2.     How many grad students are in the department?

3.     What is the average number of years it takes students to complete a PhD?

4.     Is there mentoring for teaching (such a Preparing the Professoriate Program or in-depth TA training)?

5.     How are students funded?  Teaching assistantships?  Research assistantships?

6.     Do the grad students socialize with each other?

7.     What kinds of qualifying exams are administered?  What is the pass rate?

8.     Is there good mentorship (ask the current grad students)?

9.     Where do grad students live?  What is the rent like?  How is transportation to campus?

10.  Are there awards and fellowships available?  Money for computers?

11.  What are the primary research areas in the department?  Are there strong research groups and collaborations?

12.  What is the graduation rate of students (% successfully completing the PhD Program).

13.  Where have recent graduates gone to work?

Other Useful Resources: