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?
Answer
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/27299/on-starting-graduate-school-and-common-pitfalls
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