Applied/Pure Math PhD Coursework

Question

For the purpose of curiosity, I was wondering what set of courses undergraduate math students take if they're on the track to apply for a PhD in Pure Mathematics vs. Applied Mathematics? Let me try to elaborate as best as I can. I mean at my current university, I've known several people who have gone through the track of going for a PhD in Pure Math and they have taken lots of Grad level Math courses and gotten into top US universities for a PhD in Pure Math where research experience in relevant fields is a plus (I think).

But my question is for undergraduate students trying to prepare for a PhD in Applied Math. What sort of coursework do they go through? I mean research experience (I believe) becomes important and taking graduate courses in Pure Mathematics is not so. I do not know much people who went for a PhD in applied math at my school.

I would appreciate it if anyone who can comment on the relevance of pure math grad courses for such students who aspire or strive in applying for a PhD in applied math. Any other comments pertaining to this is welcomed ;)

Answer 1

I agree with many things in posdef's answer, except that you might just get your bachelor's and apply straight to a PhD in the US (instead of getting an undergrad and master's, and then applying to a PhD). I should also say that any required course-work will vary from university to university. Many universities have some sort of 'qual' process, where you need to know certain things and pass certain tests at the start/end of your first/second (varying by university) year.

I'm a Brown PhD math student, and we have to pass our quals by the end of the first year, essentially; whereas I have a few friends at places like UChicago or Berkeley, where quals can be more immediate. From what I can tell, the subjects are almost always a subset of real analysis (and probability for applied math), complex analysis, algebra, topology, manifolds, and differential equations. I mention this because the subjects and level of testing can be very high and advanced, and if you did not do a sufficient amount of coursework in these areas, then you would probably have a very hard time passing the quals. (Really, the admissions process would probably take that into consideration, and would be less warm in the admissions process).

This is to say that there is a certain "core material" that many PhD programs seem to care about (although the exact material might vary from school to school). I would say that you absolutely must take coursework in analysis, topology, and complex analysis. But I suspect these are required courses in your studies.

But other than that, you should take classes that interest you, and apply to schools that have good programs in what you're interested in.

Answer 2

It might be somewhat controversial for some here, but I don't think there is any one course that you should take in order to get in a particular graduate program. The reason behind my statement is that I believe the logic presented in the OP is rather backwards; one usually pursues graduate studies in a particular field that s/he is knowledgeable and interested in. In other words, you get your undergrad and Masters, and based on what you know and like, you apply to PhD programs in fields where you are competent. The other way around (deciding on a PhD program without have the undergrad and masters done, and choosing courses based on the desired PhD program) does not make much sense, in my humble opinion.

Also consider that "a PhD in pure/applied maths" is really ill-defined. I can think of a hundred different projects that would have different requirements, with regards to previous courses. I would recommend deciding on a specific subject that you find interesting e.g. "elliptic curve cryptography" or "convex optimization" etc (not just pure/applied math).. then look for announced PhD programs based on projects focused on these subjects of interest.

All that being said; I would think advanced level courses in matrix theory (decompositions etc), functional and complex analysis, as well as optimization theory would be useful in many different graduate programs at most maths departments.