I wouldn't presume to answer for Terry Tao but such a foundation certainly included the usual topics covered in qualifying exams at top departments. For example, Princeton: http://web.math.princeton.edu/generals/topic.html. Now that I'm looking at it, these topics actually align fairly nicely which what Chicago requires first year graduate students to take: http://www.math.uchicago.edu/graduate/grad_first_year.html.

If you're looking for some interesting reading, Princeton actually posts write-ups of each student's qualifying exam (http://web.math.princeton.edu/generals/index.html), including Terry Tao's (http://web.math.princeton.edu/generals/tao_terence).

I wouldn't suggest focusing on qualifying exam topics unless you need to pass a math qualifying exam.

In modern mathematics, particularly applied mathematics, these are NOT the most useful topics. That's just how things have always been done, and no one wants to argue about changing them in faculty meetings. See also the language exam: http://web.math.princeton.edu/~templier/language-exam.txt

In my former career I never once used topics covered in "abstract algebra" (note that I'm distinguishing linear algebra from abstract), a common situation among analysts. Most people outside of analysis never use complex variables.

The only common core of topics I can identify that nearly every mathematician I know uses is:

Measure theoretic probability (this intersects with real analysis)

Linear Algebra

Algorithms and optimization (this is less common than the above two).

I learnt how to do proofs and what proofs really mean in my abstract algebra class. I still use abstract algebra to refresh my math skills.

I learned proofs in a numerical analysis class. That doesn't mean a budding algebraist should study numerical analysis, it means you will learn how to do proofs in any rigorous math class.

Sorry didn't mean to imply that one needs to learn abstract algebra to do proofs. Was just mentioning my experience as it was course where I really enjoyed doing proofs.

Wondering why this is voted down. He is quite right. Several math professors at my school who teach applied math do not know even the utmost basics...say things like computing kernel of a quotient group. Am sure they must have known it at some point during their student years, or maybe during the 60s-70s, you could get away without knowing these things, I dunno. otoh, my abstract algebra professor was able to learn and then teach me Ito's lemma all the way up to semimartingales, topics he had never set eyes upon in his entire life! So I'd say the abstract stuff makes you much more stronger - gives you tools to understand from scratch material you've never seen before. The applied people generally glaze over when you mention math topics out of their competence zone, especially topics that are "useless" for some definition of "use".

Do you think this actually reflects the subjects validity? I'm sure that applications can be found, but is it because people cannot do it?

I don't know what you mean by "validity". I am absolutely certain that applications of all the "standard" fields (real, complex, algebra) can be found. They aren't bad things to learn.

I'm just suggesting that from what I've seen, the other fields I listed are more commonly useful. Everyone I know, across pure and especially applied math has used probability and linear algebra. Lots of people haven't used complex analysis or algebra ever, except on the qual.

Terry was probably like 17 years old when he took his qualifying exam? Interesting quote considering his later work:

"After this, they decided to pass me, though they said that my harmonic analysis was far from satisfactory."

I'm a math professor. I like to point students to this when they have a rough time with their quals :)

I have read his summary of his qual linked here before, and upon reading it this time I also read a few additional. My takeaway is that it seems like it would be a lot of fun for the profs to conduct these interviews (most of the time, with probably a few utterly disastrous one thrown in for good measure).

Unrelated: why was your post not repliable when I first saw it? I refreshed the page a few minutes later and the reply button was there. No idea why that happened?

> a lot of fun for the profs to conduct these interviews

They can be. But it can get boring if your colleague wants to see the details of some boring computation. And, more seriously, it can be painful if the candidate is doing poorly. Fortunately I have not yet had to fail anyone.

> why was your post not repliable when I first saw it?

I believe there is a delay, the length of which is a function of how deeply nested your comment is, to encourage more top-level comments. (In particular, it tends to defuse arguments if you have to wait a long time to reply...)

One way to help get through quals: The quals are, on the surface, in some common justifications, to be more sure the student can do the dissertation research. Okay.

Well, there's another way to be "more sure", really, more reliable than any quals can ever be: Have the student do the dissertation research independently before the quals. Now in this case, just what are the quals for?

Or, for an engineering Ph.D., a guy writes a good dissertation in applied probability with careful attention to the tricky subject of measurable selection, and want to hold him back due to some qual with some tricky issue about Feller I probability?

Or, there's a qual in optimization, in part on the details of the Kuhn-Tucker conditions, but the student has already done original, clearly publishable work, e.g., in JOTA, in optimization and, in particular, the Kuhn-Tucker conditions?

E.g., for problems in functional form, are the Zangwill and Kuhn-Tucker constraint qualifications independent? Along the way, given a closed subset of R^n (usual topology), is there a function f: R^n --> R positive on the closed set, 0 otherwise, and infinitely differentiable (not quite the same as the Whitney extension theorem)? Is the Mandelbrot set closed and, thus, an example? What about a sample path of Brownian motion? So an infinitely differentiable function can have a bizarre level set? What does this say about a question, without an answer, in the famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa? And in this case, the qual in optimization serves just what purpose?

The quals might be to see if the student is prepared to take advanced grad courses, but he's already done that and used some of the best content in his Ph.D. research? Now what are the quals for?

The quals can start to obscure a basic point about the three things important in high end academics, research, research, and research, that is, the publishable kind. If a guy is doing well in research, want to hold him back because of what?

Can there be such students? Yup. I have an existence proof.

Besides, quals can be awash in politics.

The usual criteria for publication are that the work be "new, correct, and significant". At some good research universities, there is no coursework requirement for a Ph.D., and the dissertation is supposed to be "an original contribution to knowledge worthy of publication" or some such. So, for a student, do some research and publish it.

As I recall, at one time the math department at Princeton said that the grad courses were introductions to research by experts in their fields, that no courses were given for preparation for the qualifying exams, that students were expected to prepare for the quals by independent study, and students were expected to have some research underway in their first year. Good. I'd done a lot of independent study before I went to grad school.

I got accepted as a grad student at the Princeton math department but went elsewhere (where my wife was still in grad school), brought my own research problem with me to grad school, and did my dissertation research independently in my first summer building on one of the courses in my first year. Then the quals were for WTF?