What to look for in applicants to graduate programs (in mathematics)? Hello,
I was thinking about what should be looked at when deciding on the admission of applicants to graduate programs in mathematics and I thought MO would be a good place to get opinions. What do you think?
 A: The ultimate goal of the selection process is to get students that are going to succeed in the program. What this means very much depends on the program: a person who would not pass comprehensive exams in Chicago may do well in a less intense school. Also the school tries to admit students that are likely to accept; I was surprised how many admitted students in my school choose to go elsewhere. Admission is a balancing acts based on past experiences of what is a good predictor of success. 
My guess is that a big state school will first look at GRE scores (all of them including math subject). Solid math courses matter, e.g good grades in undergraduate analysis and algebra I/II, as well as in graduate courses. Nontrivial research experience is a big plus. Decent Putnam scores matter (above 20). Recommendation letters are read carefully, and what matters here is specific information, not generic praise, e.g. "student was among 3 top people in 20 student algebraic topology class that covered first 3 chapters Hatcher's book". It also matters how well-prepared and thoughtful the application is; one should think carefully what to put in the essay because it is a measure of student's maturity and good judgement.  
A: Letters of recommendation from people who have experience with graduate students. They know what it takes and can judge if the applicant is qualified.
Were you asking from the point of view of an applicant or from the point of view of someone evaluating applications?
A: I find a sense of curiosity important.  It is unfortunate that students will spend years working on a PhD before they begin their dissertation, only to discover they hate research.  While it is vital that a student can learn math that has already been done by others, intellectual exploration is necessary to continue on in research after graduation.  We emphasize the need to solve problems, but in research, it is also important to have the independence to determine what you are interested and to ask your own questions.
While this is something that does not show up in the application itself, I would find my interest piqued by a reference letter that mention's a student's curiosity.  
A: It seems that these questions always end up with answers skewed towards tier-1 research universities and especially the study of pure math rather than applied... could anyone comment on applied math or tier-2 research universities?
A: Can I ask for clarification on some of the responses?
Igor Belegradek and Paul are both very specific about which classes should be taken, and that "good grades" should be achieved. How does one define "good grades"? Does this mean all As? Mostly As with some Bs? Or perhaps As in undergraduate class with As or Bs in the graduate level classes taken?
(I believe Paul addressed the related question of what "good" GRE scores are.)
P.S. Based on my understanding, it would be more appropriate for this to be a comment on one of the answers but I can't do this without reputation points. So apologies if I am breaking the MO etiquette.
Edit: Another question: Does it make any sort of impact to have been an undergraduate grader and/or course assistant?
A: I have no experience with MS programs. The following is for American doctoral programs.
The applicant should have a decent command of spoken English. Communication is important, in class, in research discussions, and in the likely TA duty. 
The applicant needs to have a solid undergraduate background, preferably with a few graduate courses thrown in. Some graduate programs are more forgiving of inadequate backgrounds than others, but the fact is that the requirements for graduating with a mathematics major in many US schools are far below what is needed to begin graduate study. If a candidate has not taken some graduate classes in mathematics, there should be some good explanation for this, e.g, that the student changed from physics to mathematics. 
There should be some indication that the applicant knows what research is, and that the point of graduate study is research. It is to be expected that most attempts at undergraduate research in mathematics do not produce meaningful results. 
The applicant should have an area of interest which coincides with that of a professor in the department who is taking more students. This is a more important issue in smaller departments. 
High GRE scores are expected. Their absence is an indicator that something is wrong. 
It's a plus if the applicant has shown some level of dedication to some area which might not be related to mathematics, but shows discipline and perseverance. Programming skills are a plus. 
Outside funding is rare, but a plus.
I'm skeptical of the use of Putnam scores. I say that as someone who had decent Putnam scores and who coached a top 5 team. Putnam performance is correlated with ability to solve puzzles rapidly and practice. It's not as well correlated with research ability. 
A: I've done a lot of application reading over the years, although none in the last 3 years. This is a huge topic, but here a condensed (!) version of how I see it. The essay, letters, transcript, and GRE scores are the central parts of the application.


*

*On the essay: 
I discount undergraduate math research (REU) experience, a topic often emphasized in the essay. I found no correlation between a student
having participated in an REU program and their success in grad school. Dont assert that an REU gave you insight about what math research is, in 99.99% of the cases it didn't.
Don't BS in your essay. "I am currently studying aspects of the Langlands program" is unlikely to be believable except in very unusual circumstances. Honesty and enthusiasm  is what I valued most. What I want to read is "I love math and am willing to work like a slave for years to try to become a mathematician".   

*Letters:  This is a very difficult issue, since letter writing is taken seriously by some writers and,   are written by the student herself and signed by the professor in other cases. Honesty and specificity from the letter writer gives the letter more weight. Ask your professor to be clear and give specific comparisons to other students. The student should therefore make themselves familiar to their letter writers in the months before applications are due. If your professor says "Write a letter, I'll sign it" get someone else. Don't ask someone who hasn't had substantial mathematical discussions with you to write a letter.

*Transcript: Good grades in all subjects indicate a serious student.   Lots of math classes can be good, but is not always necessary. For students from the USA, strong performance in   year long Algebra and pre-measure theory Real Analysis is the most important thing, followed by good performances in other theoretical classes.  A few graduate level classes are a bonus, but not critical if there are plenty of serious math classes. 

*GRE scores:   I valued a reasonable score (at my school, a research-1 University, reasonable means top 50%) on the advanced subject portion. Unless things have changed in the last 3 years, GRE scores from mainland China were not reliable, and skewed everyone else's scores down. Perhaps the problem has now been solved and the scores are more meaningful.
A: Let me take a crack at the question, since I am currently on the graduate committee at UGA.  [The University of Georgia is about the 50th best department in the country, so just a little shy of being a research 1 university.  We are strongest in algebra, number theory, and algebraic geometry and are able to attract some excellent students in these areas.]
The two most important things for us are:
1) Very good to excellent grades, in courses which go beyond the minimum necessary for a math major and include, if possible, at least one graduate course.  We are looking for all grades B or higher and at least as many A's as B's (which implies a GPA of at least 3.5).  One or two poor grades will not concern us too much if they are in lower level courses, are followed by several years of better grades, or some explanation is given in the personal statement and/or the letters of recommmendation.  A successful applicant will probably have taken real analysis, abstract algebra and topology.  We certainly do take into consideration the student's school: e.g. a student from a liberal arts college may not have any graduate courses available.
2) High GRE scores.  We like to see at least 700 on the GRE quantitative.  It is not as criticial, but I would like to (and often do not!) see at least 600 on the GRE verbal; both scores are kept in mind by the university when it makes decisions about who will get prestigious fellowships.  As for the GRE math subject exam: I am sorry to say that as of this year we do not require it.  After looking at other universities of equal and greater status, we have decided to start requiring this exam next year (i.e., for students who are applying to start in Fall 2011), although we recognize that this may shrink our applicant pool.  A score in the top 50% on the math subject exam looks good to us.  
Next come the recommendation letters, which we use to gauge the student's enthusiasm, ability and preparedness for graduate school as compared to other aspiring graduate students.  It is much better for us if the letters come from someone that we have heard of, or whom we can verify is a successful research mathematician (I have looked some recommenders up on MathSciNet).  Good things to see in such letters are comparisons to other students who have gone on to be successful at research 1 graduate programs.  
REU experience looks good, especially if accompanied by a recommendation letter from the REU supervisor who can be specific about what is accomplished.  Sometimes students do enclose papers or preprints that are the result of REU work.  Again we like this in general (more if the paper looks interesting, less if it looks rather trivial) and may forward this along to other faculty members to see if they are especially interested in the student.
I would say that the personal statement is in fact not very important, except perhaps to address/explain weaknesses in other parts of the application.  [I accept that it might be more useful at a different institution.  I have also advised graduate students and postdocs applying for academic jobs that their cover letter is not very important, and I know that some people -- especially at liberal arts colleges -- have said exactly the opposite.]  It is useful as a writing sample, and a lack of spelling, grammatical and punctutation errors is evidence that the student is serious about their application.  
In fact it is probably more likely that you will lose points in your personal statement than gain them.  When I was a college senior, we had a Q&A session about applying to grad school.  The head of the computer science department (Lance Fortnow, I think) told us the following story: he once had an application from a candidate who had very strong grades, GRE scores and recommendation letters.  But in his personal statement he was asked "Why do you want to go to graduate school in computer science?" The candidate's response "Because I am trying to avoid working very hard" was exactly the opposite of what the department head wanted to hear.  The rest of the application was so strong that, albeit with some misgivings, this candidate was admitted.  The result was disastrous: the candidate really didn't want to do any work so was (of course!) a most unsuccessful graduate student, eventually getting kicked out of the program.  The CS department head concluded that after this experience, he would never admit a candidate who said something like that on their personal statement.  (And neither would I.)
