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".