Why do most mathematics PhD programs in United States have rigid qualifying and breadth requirements for every PhD student?

Question

For a long time, I have been confused about the purpose of having (roughly the same) qualifying requirements and breadth (course) requirements for every mathematics PhD student (in many, if not most, mathematics departments in the United States).

I have heard that European math gradaute programs do not have such rigid requirements. In U.S., although many math PhD students only start with an undergraduate degree and they do need a lot of training in graduate math, there are still a lot of students who come with a master's degree and that is the reason I add the qualification "everyone". I am just asking why we need (roughly) the same requirements for everybody in a program instead of having tailored requirements for individual PhD students, especially for those who have already taken graduate level classes, had their confirmed interests, and even written a Master level thesis in mathematics.

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In response to the comments below, I will add some parts that I deleted earlier back.

An example: many programs, such as Stanford and Ohio State, require students to take qualifying examinations in real analysis and algebra. (Some programs require exams only in real analysis and algebra, despite having strong research groups in other fields like topology.) Why are real analysis and algebra both necessary for everybody? Does an algebraic geometer have to be so familar with abstract measure theory? Does a topologist have to be so skilled in using the "Big Three" theorems of Banach spaces? Does an applied mathematician have to know Galois theory so well? Moreover, the real research is quite different from exams. The statements of problems are usually not well formulated, there is no specific time limits, the techniques can be convoluted, etc.

Taking courses are not the only way of learning things in a PhD program. PhD students can learn a subject on their own or through student reading seminars. I am not denying the necessity of taking classes for certain students (especially those who come with undergraduate degrees)---that is why I added the qualification "everyone".

Answer 1

So far as I've seen, in the U.S., the "required" aspects of most graduate programs are really very minimalistic and elementary. So it misrepresents the situation to refer to them as "rigid", since they're so minimal. And it misrepresents the situation as well to refer to those minimal bits as somehow clearly un-necessary for ... "non-specialists"?

I'd claim that these basic requirements are just to inculcate awareness of the various basics. Again, minimalist.

Further, some places have choices or other requirements. E.g., many require some complex analysis. We in MN do, as well as some algebraic topology and differential geometry... although students who want to style themselves as "applied" can manage to avoid quite a bit of this, by doing some "math modelling" and "numerical analysis" coursework.

True, in principle people can mostly learn whatever they want outside of formal courses, and I myself prefer that, because the artificial schedule of MWF classes, etc., does not fit my personal rhythms. But for some people, manifestly, this is easier than self-study. Also, I've been told by many grad students that it's easier to avoid self-deception/delusion in a more formal classroom setting. Ok.

And, of course, some people arrive in grad school being vastly over-confident of their own scholarship, as well as the significance of their "prior research experience". To make decisions based on very-incomplete information would be unfortunate.

And, of course, classes and exams are a very artificial, caricatured version of mathematics... which I think is best appreciated as not being really a "school subject" at all. Nevertheless, it is not easy to construct viable alternatives. Here in MN, anyone who really has already picked up a bit of competence in the basic subjects can "test out", by exam, and not have to take those courses. Many do. We do not try to test "mastery", but only "(at least) minimal competence".

Yes, having been involved in grad studies throughout my 35+ years here, I have often seen students complain about the alleged burdens of our "requirements". Mostly, that convinces me that they're sufficiently unaware so that they appreciate neither the minimality of our requirements, nor the universal utility of the ideas. :) But I understand that "requirements" are often viewed as "obstacles to getting on with research". I do claim that this represents a substantive misunderstanding of what's going on...

EDIT: as for a further question implied by, or suggested by, comments... although of course inertia plays some role, faculty in every grad program "require" things of grad students (regardless of eventual "specialty") after substantial consideration. Not whimsically. Indeed, exercise of such judgement is very important. The possible fact that entering grad students (arguably in possession of very little information about what they'll need later, etc.) may disagree or resent "requirements" will not deter faculty from requiring what they consider wise, based on decades of experience.

A related question, that of requiring reading competence in French and/or German, etc., has a substantially different answer. E.g., while I do still encourage my own students to learn at least French (to read Sem. Bourb. at least), and maybe German (Hecke, Siegel, ...), our program as a whole is ever-reducing these requirements, since, indeed, a greater and greater fraction of contemporary mathematics is written in English. Not all!!! I have always regretted that I didn't learn enough Russian to be able to read mathematics in it... Sure, one can "get by" being substantially illiterate, but it is quite often useful to be less illiterate! :) This applies also to awareness of the rest of mathematics, outside one's tiny bailiwick ... however large it may seem in a myopic view! :)

Answer 2

Speaking too broadly, the US and Europe have different models for PhD programs in mathematics. This reflects partly that they have different models for undergraduate programs in mathematics, and so students entering PhD programs do so with different backgrounds. It also reflects partly a difference in philosophy. (Speaking of "Europe" as a unified whole is obviously ridiculous, but for this discussion it is probably not as ridiculous as it would be in some contexts, provided one's notion of Europe does not include Britain where the educational system possibly has more elements in common with the US.)

In Europe an undergraduate studying mathematics typically takes only courses in mathematics, perhaps with some complementary courses in physics, computer science or statistics (closely related fields). In the US an undergraduate studying mathematics might take well less than half her courses in mathematics (My case is a bit extreme, but it makes the point - I got an undergraduate math degree from a US university taking 14 of 38 courses in mathematics, and 2 or 3 more in related fields, depending what one considers related; this is simply incomprehensible in most European systems). One consequence of this is that when a student finishes a European undergraduate program in math that student has typically seen a lot more math than a student who finishes a US undergraduate program in math (the situation is really far more complicated - the far greater flexibility of US undergraduate programs means that it is possible for a student to take a great deal of math, even at the graduate level, as an undergraduate, and so the distributions of courses realized are very different for US and European students - (again speaking too broady) in any given European system every student takes essentially the same classes, whereas in the US two students with math degrees can have taken radically different sets of classes). The content that the typical US student hasn't seen as an undergraduate is frontloaded onto the doctoral program in the form of the classical year long first year doctoral sequences in analysis, algebra, and something else - note that many US doctoral programs allow an incoming student with more preparation to take qualifying exams early and skip some or all of the required courses - such contingent mechanisms handle the better prepared entering students coming from Europe or Asia or from a US program that allowed the student to advance rapidly. In practice the administrative flexibility of US doctoral programs means that "requirements" are adjusted or waved as is necessary to accomodate individual circumstances (I know a successful researcher who never passed one of his qualifying exams). In Europe, "requirements" usually really are requirements in the absolutist sense of the word, and the only way to endow a doctoral program with a similar flexibility is to minimize the formal requirements.

The preceding should all be qualified by the observation that roughly half the PhDs in the mathematical sciences awarded in the US are awarded to students who hail from outside the US, probably a greater percentage at research oriented institutions. So in fact the typical student entering a US graduate program in math comes from Asia or Europe, and was prepared in a non-US system. Although this is less true in Europe, it is increasingly true in Europe as well. Nonetheless, the institutional expectations implicit in how graduate programs are structured are typically based more on the corresponding expectations implicit in how local undergraduate programs are structured than they are on the nature and origin of the students entering such programs (the latter takes longer to assimilate administratively). (Institutional expectations need not be set by mathematicians, nor even by academics; often they are imposed politically or administratively by people with little understanding of what they entail except in a formal, procedural sense.)

The other reason is philosophical. This is a personal impression, probably quite debatable, but my impression is that European doctoral programs tend to be more focused, the idea being that students immediately start research, and develop deeply in one particular area, while US doctoral programs tend to be less rushed, and have as an explicit goal that the student acquire a certain breadth, the idea being that really novel ideas often come at the interfaces between well established research areas, or by applying ideas from one context in another. For example, in Europe a student typically chooses an advisor before entering a program (i.e. enters to work with a specific person), whereas in the US the choice is typically made in the first year or two of the doctoral program. The US structure allows the student to explore her interests for a while before focusing; the European model obligates focus before the interests are developed. In Europe a student is formally expected to finish a doctorate in 3 or 4 years. In the US, 5 years is normal and 6 is fine. (If we are talking about star students at "top" places all the expectations change, but I am focusing on the "typical" students who will mostly become ordinary mediocre researchers.)

To some extent the differences in the structure of doctoral programs also reflect differences in the expectations for the postdoctoral phase. A "postdoc" in mathematics in the US means a temporary teaching position during which the person is expected to develop an autonomous research program. A recent graduate needs to be prepared for that level of autonomy. In Europe a "postdoc" in mathematics typically means a position directly connected to a particular research group, in which the student is expected to participate, sometimes in a still directed manner. A student entering such a position needs less autonomy a priori than a student entering a US style postdoctoral position, but may need considerable area specific knowledge to function usefully in an established research group.

Surely some of the characterizations in the preceding are wrong in the details, or when compared with any specific context (particularly vis-a-vis Europe), but what I think is valid is the claim that the structure of doctoral programs reflects ideas about the preparation of students entering such programs and the needs of students exiting such programs, and that such expectations are different in the US and Europe, even if it is hard to articulate exactly how without oversimplifying (the discussion includes dozens of national systems ...) Also the intellectual traditions in the US and Europe are different, and this effects the structure of doctoral programs. Again, a properly nuanced examination of what the differences are would require a book, but the general feature that seems most salient is the relative importance of breadth versus depth. Maybe also the differences reflect different ideas about the importance of developing exceptional talent and training/forming ordinary talent. To me the US seems more oriented towards the former, Europe more oriented towards the latter (this does not claim that one is more succesful than the other in either regard; intention does not imply results).

Answer 3

In the US tradition, a PhD does not simply indicate research ability; it also indicates a basic ability to teach all undergraduate courses (except perhaps upper-level electives) in the field. There are many colleges in the US with only 2 or 3 math professors (in fact collectively these places hire more math PhDs into permanent positions than research universities), and a small department relies on all its members being able to teach all its courses. (If the algebraist in the department has a heart attack, the applied mathematician will have to cover the undergraduate algebra course.)

In particular, this is why algebra and analysis are almost always part of the requirements; these courses are central to the undergraduate curriculum almost everywhere.