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In short: Why don't we have the answer as "just the topics in the GRE" simply because the GRE is meant for applicants to average pure math PhD programs in the US, or is it not? if more than half of said PhD programs required algebraic geometry, then we expect the GRE to include algebraic geometry, don't we?

After the basics of topology, abstract algebra and complex analysis and number theory, what are essential topics for the average pure math US PhD program?

If there are none, then how do you know?

If there are, what are they?

  • I think they are the basics of the following topics:

    • Algebraic topology, such as Part II of Munkres Topology

    • Algebraic geometry and commutative algebra, such as the rest of the second half of Artin Algebra

    • More group theory, such as the rest of the first half of Artin Algebra

    • The algebra topics in Dummit Foote Abstract Algebra that are not in Artin Algebra

    • Differential geometry, such as Tu Manifolds

  • What about basics of the following topics?

    • Functional analysis

    • Advanced real analysis (the one with measure, Lebesgue integration, Radon-Nikodym, etc)

    • Partial differential equations

    • Measure theory

    • Differential topology

My context: I was recently rejected for a pure math PhD program for not having a strong enough background in "essential topics"; am wondering whether my background is more similar to an applicant to a US-style grad school than to a European-style grad school, but I am not asking about this. You can see the previous versions for the details. I am in Country A.

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    Why should this question stay open when your last one was closed for being opinion-based? – Bryan Krause Dec 12 '18 at 21:13
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    The Math Subject test is not at all a test of preparedness for graduate study. It's a test of preparedness to TA calculus. Part of the reason is that US graduate programs vary greatly by level. The average person earning a PhD in my mathematics program knows less mathematics than the average first-year graduate student at Princeton. – Alexander Woo Dec 12 '18 at 21:15
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    I certainly think if you don't mean to talk about the GRE you should not mention it. – Bryan Krause Dec 12 '18 at 21:17
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    My statement about "US graduate programs" is true whether Masters programs are included or not. I do mean pure math. – Alexander Woo Dec 12 '18 at 21:18
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    The question is becoming a moving target. Let it settle, please. – Buffy Dec 12 '18 at 21:24
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These questions are essentially unanswerable, because universities in the US are quite diverse in their expectations.

The average person earning a PhD from the department I work at knows less mathematics (and is certainly less capable of doing research) than the average first year graduate student at Princeton.

The average person earning a BA/BS with a major from the department I work at knows less mathematics than the average junior major where I was an undergraduate.

The average person entering my university as an undergraduate knows less mathematics than I did when I was 14 (and starting high school).

Of course these are averages and there are exceptions.

Do you want the answers for my university, or for, say, the University of Minnesota, or for Princeton?

(And, as for the ETS, they mostly are catering to be accurate for the middle of the normal distribution, which means universities like mine, because that's where the people are.)

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  • Alexander Woo, do you disagree with Buffy? Thank you! – Jack Bauer Dec 12 '18 at 22:43
  • I edited my question. Is your answer mostly the same? Also, how about non-ivy leagues? – Jack Bauer Dec 16 '18 at 11:09
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    Yes; my answer is still the same. There is no such thing as an "average" US university. By the standards of your Professor A, my university is probably not a "reasonable university". But "unreasonable universities" exist and are indeed quite common. – Alexander Woo Dec 17 '18 at 22:41
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    @JackBauer: How am I supposed to give you an answer then? Say that 19.6% of the programs expect you to know commutative algebra? And what do you mean by "expect"? Certainly at a highly selective program, those who know more mathematics will have an advantage in admissions, but someone who shows exceptional promise in other ways might be treated differently. – Alexander Woo Dec 19 '18 at 18:38
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    @JackBauer: Ed Witten's undergraduate degree was in history. Presumably, the Princeton physics faculty saw some reason to admit him to their PhD program despite his having studied far less physics than usual for a physics PhD student. Clearly they were right. – Alexander Woo Dec 19 '18 at 18:43
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If you want to know why ETS does something, ask them. The test needs to be broad enough so that most UG curricula are well covered, but not much broader than that. It tries not to disadvantage anyone in its coverage, though most students will find questions there that are about things they haven't studied. And you can still do extremely well even leaving some questions unanswered.

On the other hand, anything that you study will give you some additional mathematical background before you start to dive deep into a research area. You probably don't have time for all of them, so just choose something that seems interesting. If you already have a research interest, you could start there.

I'll note that the last time that it was possible for a single person to know all of mathematics was early in the 20th century. It has expanded too much since for it to still be possible as it once was.

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    Most UG programs cover the basics and one or two additional things. I studied PDE, for example, as an UG (half a century ago). But measure theory was an early grad course. I'm not sure that ETS makes a big distinction between MS and PhD. Many students see the MS as just a stepping stone in any case. – Buffy Dec 12 '18 at 21:07
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    I studied algebraic topology as a grad student. In fact my first serious topology course was in grad school. The UG version was a pale shadow. Nice historical emphasis (Koenigsberg Bridges, etc) but not very deep. But Analysis (real and maybe complex) is pretty fundamental and many UG programs are, I suspect, pretty heavy in that. – Buffy Dec 12 '18 at 21:16
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    Correct. Note that a doctorate in the US normally includes quite a lot of coursework. It is probably not like UK where the UG is more concentrated, so more maths are covered. – Buffy Dec 12 '18 at 22:30
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    "Is algebraic topology expected of the average pure math US PhD program under the supervision of a professor researching in topology or geometry?" Probably not if the department is weak in algebraic topology and the professor is in set-theoretic topology. I think you're putting too much focus on specific subjects. If you're simply interested in admission (and not trying to conduct a survey of existing or exiting students from graduate programs), then future potential (can you pass the qualifying exams and complete a dissertation?) and past achievement (e.g. top 20 Putnam?) play a huge role. – Dave L Renfro Dec 13 '18 at 12:01
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    @JackBauer stop changing the goal posts... You should have read about how to ask good questions.... – Solar Mike Dec 16 '18 at 11:27

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