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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?

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Would you mind explaining the motivation for this question? – François G. Dorais Feb 19 '10 at 23:40
I want to know what other matheamticians find important. – commonname Feb 20 '10 at 0:29
The Committee usually begins by eliminating all applicatns with gratuitous spelling errors. – Tom LaGatta Feb 20 '10 at 1:25
@Tom: I made sure to make an ample amount of spelling & grammatical mistakes in all of my applications; I still got in. (i.e this sort of stuff doesn't seem to matter). But are you on some committee by any chance? I'm just a grad student. – maks Feb 20 '10 at 2:12

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.

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I'm glad someone is expressing skepticism about the Putnam! As Kiran Kedlaya once put it, all it tells you is that if you enjoy problems that take hours instead of minutes to solve, perhaps - perhaps - you'd enjoy problems that take months to solve instead of hours. – Qiaochu Yuan Feb 20 '10 at 6:06
Which is not to say that there isn't a difference between a score of zero and a score of thirty. But if anyone is making significant distinctions past that then they are valuing someone who thinks and writes quickly over someone who can do research. (Full disclosure: my own Putnam scores are decent, but not spectacular. I often overestimate problems and write slowly, neither of which, I think, correlate with my potential to do research.) – Qiaochu Yuan Feb 20 '10 at 6:10
"The applicant should have an area of interest which coincides with that of a professor in the department". I don't agree with this, and this view I think is a cultural difference between the US and elsewhere, particularly Europe. At most US universities, it is not necessary to have already specialized. Indeed, I sometimes receive inquires from foreign students who want to come and work with me, and who want me to commit to this, but I am reluctant to agree before seeing the student for a year or more. Let the student come and explore his or her interests with various professors. – Joel David Hamkins Feb 20 '10 at 6:16
@Joel: Ideally, students will be flexible, and there will be several choices. In practice, some students will have specialized or will know that they are more inclined toward number theory, or more toward statistics, or differential geometry, and then it's a clear mismatch if there are no professors in that area. It's pretty bad if there are professors in that area but they are not happily taking students, which may be known in the department but not obvious to applicants. I recommend that students who are not specialized try to apply to schools with larger departments. – Douglas Zare Feb 20 '10 at 7:17
I want to mention another Putnam story, about why my college's team didn't do well one year. One of my teammates loved one of the problems in the morning. He spent almost the whole 3 hours on it. Ok, that's not too uncommon, but definitely something I taught people to avoid as a coach. What was remarkable was that he also spent the afternoon session working on it, when he wouldn't get credit for solving it. Not surprisingly, he received a poor score on the exam, but I believe he went on to a decent graduate school in mathematics. – Douglas Zare Feb 22 '10 at 18:37

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.

  1. 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".

  2. 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.

  3. 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.

  4. 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.

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@Paul, wondering what one should say about their "research experience", I say that having had some "research experience" which I am well aware is not the real deal. I don't want to sell it short but I also don't want to say something similar to "I know I will like research because this gave me a feel for it..." Any suggestions? Reason: I'll be writing apps for myself in the fall, figured I'd ask. – Michael Hoffman Feb 22 '10 at 2:47
@Paul: as someone on the graduate commitee of the UGA math department, I am interested in your statement about the unreliability of the GRE scores from mainland China. Would you be willing to elaborate on this? If the subject is too sensitive to post here, perhaps you would be willing to contact me via email. (My address appears on my user page and is anyway otherwise easy to find.) – Pete L. Clark Feb 22 '10 at 3:17
@Michael: I'm not suggesting that you should avoid writing about what might have been a very good/inspiring mathematical experience. If the REU experience drew you in and convinced you to go on in Math, say that. My comment was meant to suggest that REU by themselves don't correlate (in my andecdotal experience, anyway) with success. @Pete: I'll email you. – Paul Feb 22 '10 at 14:02
@Paul, fair enough, understood – Michael Hoffman Feb 22 '10 at 22:27
@Paul: I am a master student in a Canadian university. If I wish to pursue my PhD in a US university, do I have to take GRE? Many universities require both general test and subject test, but I don't like to take general test. Forgive me if this question is not suitable here. – Sunni Mar 6 '10 at 4:55

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.)

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"the country" being the United States. Apologies for my jingoism. – Pete L. Clark Feb 22 '10 at 3:58
:-) I somehow guessed that when you called the University of Georgia "about the 50th best in the country", the country wasn't Georgia. – Tom Leinster Feb 22 '10 at 23:34
Thanks Pete for your very insightful comment. In various colleges like mine there is no GPA system. One gets raw marks. Here pass marks are 50% and scoring 75% is really hard. And even if colleges have GPA systems, various colleges across the world have different systems. Like some colleges I know have 10 to 2 systems with A being 10 and AB being 9 and B being 8. Some have A being 10 and then next is B being 8 etc. In such cases how do you cross-compare between various marking and grading schemes across the universities around the world? – Anirbit Apr 3 '10 at 7:35
Grades are WAY overrated and I wish people would start to actually use that fact in their evaluations. – The Mathemagician Jun 28 '10 at 4:10

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.

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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?

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My guess is that the OP is asking from the point of view of the evaluators, since she asks what "should be looked at" rather than what admissions committees actually do look at. But more motivation and explanation from the OP would certainly be helpful. – Pete L. Clark Feb 20 '10 at 6:09
I had the impression this is an applicant who is phrasing the question in an ambiguous way on purpose. – Felipe Voloch Feb 20 '10 at 14:12
Felipe, you sound like there is something wrong with applicant's desire to know more about the admission process. Since your comment got lots of votes, I am curious how this is different from trying to learn more of job search process? It is no different! We should be completely open about what it takes to succeed. There is really no way to game the system in any significant way. – Igor Belegradek Feb 20 '10 at 20:47
Igor, there is nothing wrong with the question and I gave an honest answer. I'd like an honest answer to my question in return. – Felipe Voloch Feb 20 '10 at 21:29

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.

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Many (most?) people who get a PhD will not be doing independent research later. I think the goal of graduate school admission is to admit people who can get an advanced degree. In my experience an absolute majority of letters mention student's curiosity. Even if there is nothing else to mention, they talk about curiosity. – Igor Belegradek Feb 22 '10 at 17:48

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?

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This is a comment rather than an answer. But it is a good comment! – Pete L. Clark Feb 22 '10 at 3:47
P.S.: My university is (alas and alack) a research 2 university. And my comments apply equally well to applied math, which is part of our program. – Pete L. Clark Feb 22 '10 at 3:54
My comments apply to research 2 universities as well, and I see no difference in how applied math applicants are treated on admission. Everyone still has to pass the comps in algebra and analysis, no matter how applied you are. – Igor Belegradek Feb 22 '10 at 16:31
@Igor Belegradek: At some universities, students in applied math have to pass exams in applied math, not pure math. (I speak for a top-10 program in applied math.) – aorq Feb 22 '10 at 18:55
Charles, this is not how it works in my place, where by the way the applied math group is quite prominent. I certainly did not mean to imply that this is happening everywhere: my experiences are limited. – Igor Belegradek Feb 23 '10 at 3:23

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?

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@RH: By being a grader and/or course assistant, you might get to know a faculty member better and thus get a more insightful recommendation letter. Other than that, it would make little or no impact on me, at least. – Pete L. Clark Feb 22 '10 at 4:39
Being a grader or course assistant for an advanced (i.e., more advanced than calculus and differential equations) course is probably a good way to build your mathematical maturity, and as such might help you actually be successful in graduate school (though not as much as simply working hard in the classes in which you're a student). But as far as admission to grad school goes, I agree with Pete. – Mark Meckes Feb 22 '10 at 14:21
@RH: having anything below B on advanced courses, especially those in analysis and algebra spells trouble. Of course, the higher the grades the better, but the admission committee looks into how the grades were earned and what they mean. For example in some advanced graduate classes everybody who comes gets an A, so that does not always count as much. But if your letter writer describes how you got the grade, and says what was covered, that that the homework was pretty hard, this speaks volumes. The more specific the letter are, the better. – Igor Belegradek Feb 22 '10 at 16:47
@RH, continued: It is also common knowledge that some undergraduate universities are more rigorous than others, and in many cases the the committee members know the letters writes personally. Sometimes a phone call to letter writer is made to get more information. If you have had a bad semester with poor grades due to health/personal issues it may be worth explaining what happened. – Igor Belegradek Feb 22 '10 at 16:47

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