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



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".
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! :)