I am an undergraduate (currently a sophomore) studying to prepare for applying to a Ph.D. program in mathematics. I have thus far structured my course selection upon the advice of a friend I met during freshman year (he was a senior) who has gone on to a Ph.D. program himself. A large part of his advice was what a Ph.D. program in mathematics would expect from a strong applicant. Or perhaps "expect" is not the right word – his words were that having certain topics under my belt by the end of undergraduate study would make me a pretty strong candidate.
His suggested topics for study were: real analysis at least at the level of baby Rudin, preferably further real analysis (measure theory, functional analysis, etc. – probably Folland or similar), point-set topology, complex analysis, Fourier analysis, differential geometry, probability, linear algebra, and abstract algebra (e.g. Artin, Dummit & Foote). He also suggested that if I had time for it, it would be worthwhile to know commutative algebra, algebraic topology, axiomatic set theory, number theory, and mathematical logic, though these wouldn't be strictly necessary.
The notable omissions from this list, I think, are elementary partial differential equations, combinatorics, and graph theory. At the time I spoke to him, I had already done up to multivariable calculus and was doing ordinary differential equations, so those were omitted for that reason. The reason for leaving out PDEs was apparently because it would be more worthwhile to study a lot more background material in analysis and topology before moving onto PDEs, as opposed to jumping into PDEs with just ODE knowledge. As for discrete mathematics, apparently these are more optional than things like set theory and number theory – again, nice if you know these things, but not catastrophic if you don't for Ph.D. admissions.
Here is my question: is this an accurate picture of what a sample strong applicant for a Ph.D. program might look like? Are there other topics I should consider studying? I realize that students apply to Ph.D. programs from all mathematical backgrounds, and faculty recommendations, GRE scores, summer REUs/programs, extracurricular mathematical activity (seminar talks, etc.) also contribute heavily. But I imagine that there is a basic level of coursework that in addition to having good recommendations, REUs, etc. would suggest that an applicant is ready for graduate work.
My concern with this picture of recommended coursework is that the friend who suggested it to me is primarily interested in analysis, and I think it's worthwhile to have perspectives from mathematicians interested in algebra, discrete mathematics, etc. on this matter. It would also be great to have some perspectives from professors on graduate admissions committees and professors that often mentor graduate students/teach graduate classes.
If it is relevant to this discussion: right now I am leaning toward studying topology and geometry, though I am also interested in the nonlinear analysis I am reading right now. Take this with a grain of salt – I haven't studied many of the topics in the above list yet. It's very likely my opinion will change once I know abstract algebra. My current progress on the list: I am studying real analysis using baby Rudin, I have studied topology, probability, linear algebra, and combinatorics, this semester I will start differential geometry and complex analysis, and I plan to go over topology again.
Best Answer
I'd suggest you visit the American Mathematical Society's site for students: I've linked you to their page for information that addresses concerns of current undergrad students. There are links to graduate programs (and their requirements and contact information), and information about undergraduate research programs, funding sources, career prospects, etc.
Survey a few programs, their requirements for admission, suggestions to applicants, funding availability, program requirements, etc. Requirements for admission will vary from program to program, but you'll find a common basis with which to answer your own question.