Clarifications on American graduate school for an European student

Question

My advisor strongly encouraged me to check out two (highly renowned) American universities for my PhD study.

Her collaborators in these two institutions are good people and great mathematicians, so, assuming I can get in, working there would be very pleasant.

However, I'm confused by the structure of these graduate schools. In the first two years I'm supposed to follow courses and do a comprehensive written exam and an oral one which are mostly about topics (complex analysis, basic functional analysis, ODEs) that we do in the first two years of undergraduate study in my current institution.

The exam questions from the past years appear to be quite difficult, but revising those relatively elementary topics and spending a great amount of time solving difficult problems on them seems like taking a step back after a Bachelor's and a Master's degree (which had a quite significant research component).

So my questions are the following:

1. what is the rationale behind this structure of graduate school in the US?

2. why is it effective?

3. should I be concerned about "wasting time" revising basic topics in my area instead of diving directly into a research program after a Master's degree?

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Added context from comments:

"Students who already know the material can take the exams in the first month of their Ph.D. program" is exactly what my advisor's collaborators told me. However, although the core material is well-known to me, it appears that the exam consists of many problems in a short amount of time and that such problems are mostly about clever ways to sum series, evaluate multiple integrals, do contour integration, solve tricky ODEs, and so on. That is, it is about elementary things but requires lots of exercise. That's why I'm concerned that it could be an unnecessary detour.

Answer 1

1. The rationale is because graduate programs in the US are generally structured assuming the incoming student does not have a masters, and got a bachelors in the US, which can sometimes involve an embarrassingly small amount of math (and maybe more importantly, there is a very wide variation in what students learn at different universities, due to the flexibility of the US system). So, you're entering the system at a drastically different level of preparation from most people. Of course, it seems off to you. My experience at quite a number of US schools is that almost all incoming students need some sort of preparation of this type, often even ones with credentials suggesting that they shouldn't. I think some of the logic here is that it's easier to set out somewhat tougher requirements and then make exceptions to let people accelerate than the other way around.

2. I think it's hard to assess whether it really is. But I don't know anyone who seriously suggests we don't need to do something along these lines. If something is covered in the syllabus of these classes, it is because the students don't consistently come into the program knowing it.

3. Maybe I'm being too dismissive here, but I am never sympathetic to these complaints. If you really know the subject, the classes will not take much of your time. No one will stop you from talking to professors about research or doing reading on the side while you're taking the classes, and generally the timetables around exams and classes are flexible. That time will only be wasted if you decide you want to waste it.

Answer 2

Speaking as a long-time graduate program director and adviser in a mid-level US department (at a big public university), I'd emphasize that most of our Ph.D. students in past years have had a less-than-adequate preparation in some basic areas. This is true especially for US students coming from a typical 4-year college math major but is also true for some international students. On the other hand, most of our students aim for a college teaching career, or perhaps a non-academic job, rather than aiming for a research career in some kind of mathematics (or statistics, part of our department). At the most elite departments, it is taken for granted that everybody plans to do research. But my own contemporaries in the 1960s Ph.D program at Yale actually went in many different directions.

Like most other US departments, we have had an evolving procedure involving written (and perhaps oral) qualifying exams. Anyone applying to one of the top few programs should certainly ask questions about how and when such exams are administered, and how they are evaluated. Typically no one wants to hold back a talented student, but sometimes a student overestimates his or her own talent and knowledge (and future job prospects). Even Harvard and MIT Ph.D.'s sometimes end up teaching at small colleges or out-of-the-way universities.

In any case, identifying a potential thesis adviser (or two) is equally important, though obviously it's difficult to predict one's future interests precisely or to predict the future logistics of an active faculty member (sabbaticals and other leaves can upset plans as can personal crises). Good luck navigating the US system, including the evolving immigration rules!

Answer 3

In US programs I know about, e.g. UC Berkeley, there are preliminary exams (typically taken after the 1st year), on a broad range of topics, and qualifying exams, on the intended area of the PhD. Unless you went to some very, very good MSc programme, this is meant to prepare you for academic career, and broaden your view of the subject.

I wish I went to one of these, instead of diving headfirst into PhD research.

As to "why is it effective", consider excellent products the system has been producing :-)

Answer 4

Supplementing the other good answers, and echo-ing one of @BenWebster's points: I, too, am unpersuaded (statistically...) by complaints that "prelim exams" are too speed-oriented and/or too low-level/cleverness oriented. True, these are easy accusations to make, and complicated to attempt to refute.

However (without claiming anything about the questioner's situation), almost without exception, the on-the-ground complaints I've heard in this direction were from people who were not nearly as competent or mature as they imagined... and easily stood to profit greatly by further attention to basic things.

In particular (although, yes, there certainly can be "bad" instances of various prelim exams), an important meta-point that I (in writing such exams) try to test for is a higher-level understanding, especially which enables examinees to "see the forest for the trees" (or vice-versa), and realize that the given question is "just another" instance of an iconic, cliched, and useful/important idea... perhaps manifest in a slightly prankish-seeming context.

What I tell my own people is that they "are done" with "algebra, complex, real, ..." or whatever labels these exams are given, when they realize that there are "only eleven (or pick your favorite smallish-number) different issues that will come up". Prior to some point, they cannot see the commonalities (which is not so good), and then after some point, they do see the commonalities, and everything is easy.

In the latter context, almost all the complaints I've received are from people in a state prior to seeing how simple these basic parts of mathematics are (or can be). This is not a moral failing, but it is not a useful professional state (especially as part of a belief system...).

Answer 5

I can offer a more cynical point of view, as someone who has studied for and passed exams three times, at three different U.S. grad programs in mathematics: I completely agree with you. If you think preparing for exams is a total waste of time for you, you're probably right.

There might be students who actually learn things in the course of preparing for an exam, but there are also students for whom the whole exam routine actively interferes with meaningful learning. I am solidly in the second category, so I minimize the cognitive resources I allocate to exam preparation because it diverts time and energy from doing something useful. As another reply here points out, the exams at each school are usually recycled variations of the same manageably-sized bag of tricks, so it's probably less time-consuming than it seems at the outset.

From a philosophical/pedagogical point of view, it's unfortunate that programs don't even try to be a little more data-driven about the effectiveness and efficiency of this hidebound process. I could give a very long critique detailing why the coursework-written exam-oral exam gantlet is completely counterproductive, starting with the educational research that high-stakes exams reduce long-term retention. But this is the sort of thing that no one has any interest in hearing because they have no interest in doing anything about it. As you can see from the other replies, most (not all!) professors within this system are under the impression that such exams are a reliable way to determine whether a student "really knows the material," so why fix what (to them) isn't broken?