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.

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

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!

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.

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?