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I am a second year under-graduate student in mathematics. My school (3 years under-grad) has a great history of sending its under-graduates to top graduate level math programs of the US universities. However, I do not know to what extent top universities expect an under-graduate to have done graduate level courses.

The math courses I will have done in my under-grad are: Calculus I-II-III, Analysis I, Analysis II, Linear Algebra I, Linear Algebra II, Complex Analysis, Real Analysis, Intro to Algebra, Discrete Math, Introductory PDE, Stochastic Processes. I do not know if I will have done Topology and Number Theory as they are not offered in a regular basis.

So, my question to those who are studying or working in graduate level math departments is: Are there any specific (number or name of) graduate level courses that a student should have done to be accepted in top math schools for PhD? I know that many things other than the transcript would be considered by admission committee. But just looking at the transcript of an undergraduate, which graduate level math courses would the committee like to see in applicant's transcript?

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Disclaimer: This is all vicarious based on professional mathematicians' advice (from MathOverflow and the like). Take these words with a grain of salt. . .

Graduate schools (especially the "top-tier" ones) judge more based on your ability to do research rather than your ability to get an "A" in a class or the quantitative aspect of your CV (referring to how many graduate-level classes you took). In other words, what matters most is your potential ability to become a producer of mathematics, rather than a consumer. After all, your PhD thesis won't come out from that (necessarily). So here's what I think you should be focusing on at the moment, for your final year (not in order of importance):

1) Getting glowing letters of recommendation (at most 3) from professors who know you well. They'll probably be able to accurately asses your research potential assuming you've done some type of research project with them. If you have not, get to it. The key is to develop a close relationship with someone (or a couple of people) in your department who'll be able to get a good word for your potential as a research mathematician. Try to look for someone whose area of research coincides with your potential speciality in grad school.

2) Start on a research project (independently or otherwise) if you have not already ASAP. Something like a senior thesis. Research that result in publications look awesome but are definitely not necessary (I don't think anybody expects much out of undergrads). As long as admission officers can say "Hey, this guy/gal is motivated enough to do research on their own, and hey look at that work they're doing, there might be hope for them yet."

3) If you're planning to apply to places like Princeton, high GRE scores are preferred (shoot for the 90th percentile and above). I hate to think of the admission process as a bureaucracy but just put up with it. Just remember, these exams are definitely important, but they are only one part of your application to be considered.

4) Challenge yourself, and keep your GPA optimized (I do not mean taking "easy" courses that just fluff up your GPA. Try to take the relevant math, which probably means at the upper or graduate-level). Though coursework at this point should not be your main concern (research should be), if you think you can handle it, and you think you'll have time for it, take a few graduate-level courses. Also, fill in the topology gap. Graduate admissions will look for a solid undergraduate background in mathematics (real analysis, abstract algebra, and topology are probably the "Big 3" for an undergrad to take, in addition to other classes). note: the fourth point seems repetitive now that I'm re-reading but I'll keep it like this for now for informative purposes.

Other qualities that'll help (more tongue-in-cheek than actual advice) is the ability to persevere and have genuine curiosity for whatever field you're going in when writing your personal statement. Do not puff up your application with courses just to "look good". General knowledge helps, but this is something you can do on your own time. Focus on optimizing your GPA and standarized test scores (GRE); most importantly, focus on establishing a professional relationship with a professor (letters of recommendation) and research. Other than that, good luck to you.

You may also want to browse these threads. I've pretty much reiterated most of the points, but you should still see their different points of view.

https://mathoverflow.net/questions/15848/what-to-look-for-in-applicants-to-graduate-programs-in-mathematics

https://mathoverflow.net/questions/27299/on-starting-graduate-school-and-common-pitfalls

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    keep your GPA optimized — But remember that in this context, "optimized" is not a synonym for "maximized". A+'s in easy undergrad courses are worse than A-'s in killer grad courses. – JeffE Jul 4 '13 at 7:37
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I am a professor in a top ten math dept. in the US. I have sat on admissions committees for many years, and talked to colleagues at other top tier institutions about admissions.

First: admissions is not done by a "bureacracy", or "officers". It is done by some subset of the math dept. faculty, who read the applications, including the letters, the GRE scores, the transcripts, and the essay, and then rank order them.

Second: To get to a top institution, high GRE scores are essential. These don't guarantee admission by any means, but if someone can't get high scores on this exam, it calls into question their understanding of all the basic undergrad math they've learnt. It's true that different schools (and different faculty members, even at a given school) place different weights on this exam, but doing well on it is something you have some control over in the admissions process (by preparing well), so it makes sense to do so.

Your GPA in your math courses is also very important. Presumably you are doing your best in your courses, and getting as high a GPA as possible. So there is no real magic to this; you just have to work hard at learning math.

Third: Assuming that your GPA and GRE scores put you in the ball-park of being a credible candidate for admission, people will read your letters carefully. So you want to get letter writers who can write about your achievements and abilities in as much detail as possible.

Fourth: No-one expects undergrads to have done real research; REU experience certainly helps, but one main reason for this is that it provides a way to meet professors who may get to know you quite well, and so can (hopefully) write a strong letter for you.

Fifth: Graduate courses certainly help, if your grade in them is good and meaningful. It is often the case that undergrads in grad courses will be given somewhat inflated grades out of sympathy on the part of the instructor. This makes sense from the point of view of not destroying someone's GPA because they took a challenging class, but an admissions committee will look for evidence that the undergrad really did master the material in the grad classes they have taken. One way to show this is to have the instructor of the graduate class write a (hopefully positive) letter.

Again, different schools have different expectations about what incoming students will know. At the absolute top places (Harvard, Princeton) essentially all the incoming students will know essentially all the material in basic graduate courses (measure theory and functional analysis, basic algebraic topology, basic commutative algebra, and so on). At other places this is not the case, but most incoming students at most top schools will be familiar with a reasonable percentage of what would be regarded as basic graduate math.

Sixth: In terms of preparation for grad school, writing a senior thesis is great. It teaches you about a topic in much more depth than you would normally investigate it, normally leads to learning some grad level math, and also again builds a closer connection with a professor (your advisor) who can then hopefully write a good letter for you. And at a more fundamental level, since mathematicians spend most of their time writing about mathematics, this is pretty good preparation for that.

As far as applications to grad school go (rather than as preparation for succeeding in graduate school, where as I've said it's great), my sense is that the real pay-off for writing a senior thesis is the letter from the advisor.

Since this is a bit harder to quantify than entries on a transcript, often students are advised to take grad classes rather than write a senior thesis (if this option exists). I can see why this advice is given, but I do think writing a senior thesis (if you put your heart into it) is an invaluable mathematical experience.

A related option is to do independent study on a topic with a faculty member. Again, from the pragmatic point of view this doesn't do much for your transcript, but the benefits for your mathematical education are similar to those of writing a senior thesis, if you have a good advisor and take the independent study seriously.

Finally: Remember that there are lots graduate schools in mathematics. What I've written above more-or-less reflects admissions experience at top ten institutions, but obviously there are many more very good math programs out there, and they are not all as competitive as the top ten.

So overall, the most sensible thing to do is to work on learning as much math as you can as well as you can, and interacting with your professors enough that they know what you're learning and can write about it. I think the only things that are really worth thinking about in terms of "gaming" the admissions process are attending an REU or doing independent study or writing a senior thesis, something that gives you a chance to interact with a professor, and a specific topic, in more depth; and making sure you study well for the GRE before you take it.

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    This agrees with my experiences as a spectator of the admissions process / math grad student. – davidlowryduda Sep 12 '13 at 4:33

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