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Edit: Right so I forgot this really big thing. My question is actually under this framework: Why are US PhDs different from European PhDs?

  1. After the basics of topology, linear algebra, abstract algebra, elementary analysis and complex analysis, what are essential topics for the average pure math US PhD program?

  2. If there are none, then how do you know? If there are, what are they?

  • Guess if there are: 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

    • Elementary differential geometry, such as Tu Manifolds

  • Guess as to what isn't included:

    • basics of complex geometry

My context: (Feel free to just ignore this bottom part if it makes my post too long.)

  1. I was recently rejected for a pure math PhD program in Country A, where I live, for not having a strong enough background in "essential topics".

    • 1a. The professor said I was not ready for a PhD or even an MPhil in pure math. I asked if he meant for Country A (I was very careful to not use the words "only" or "just"). He claimed that it was for all "reasonable" universities.

    • 1b. He claimed that the average first year Country A PhD student (before starting the programme) in topology or geometry would know the basics of algebraic topology, complex geometry, Riemannian geometry and more algebra than the elementary abstract algebra. Some of the specific concepts are Gauss-Bonnet, branched coverings, Kähler manifolds, Poincaré duality, Euler characteristic, etc. Also, there's stuff about Riemann surfaces required (eg Mittag-Leffler and Riemann-Roch. I'm guessing also Abel-Jacobi, Riemann-Hurwitz, Poincare-Hopf and the list of 2-name concepts goes on.)

  2. I am wondering whether my background is more similar to an applicant to a US-style grad school than to a European-style grad school.

    • 2a. Consider Johns Hopkins University. Its maths phd requirements are the same as the "straight phd" programs in the top 3 universities in Country B, where I'm from and where I got my bachelor's and master's degrees in (unfortunately applied) math. (These 3 universities have "regular phd" which require master's or equivalent and "straight phd" which require only bachelor's or equivalent.) Both JHU and top 3 universities in Country B are actually even less than what I put as my 'guess' above.

    • 2b. But anyway going back to JHU, it even says 'Nevertheless, the department does admit very promising students whose preparation falls a little short of the above model.' In other words, JHU isn't even as strict about these elementary requirements, but this Country A university is extremely strict about these highly advanced requirements.

    • 2c. I just find it very hard to believe that my rejection from this Country A university isn't related to these US vs European questions that I've asked before. I would like to think that my rejection is that European universities simply require more. I don't quite have a chance there without further studies, but I do have a chance in the US. (And worst comes to worst, there's always Country B.)

  3. You can see the previous revisions for more details.

Gonna copy some comments into the post:

  1. If you're fact-checking this professor, what he said is kind of stupid. If you're trying to assess your preparation for whatever an "average" program is, you need to check with them. – Elizabeth Henning Dec 17 '18 at 16:41

    • (I think this comment is about the "reasonable" thing.)

  2. I honestly don't understand what you're after here. All programs will say they want an undergraduate major in math or a related field. (...) A mid-ranked school will probably expect basic abstract algebra and calculus with proofs. (...) – Elizabeth Henning Dec 19 '18 at 16:57

  3. (...) It is very likely true that more than half require no more than the GRE topics, if that is useful information. – Elizabeth Henning Dec 20 '18 at 4:52

BCLC
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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
  • This question was voted to be closed as primarily opinion-based. I think (2) is opinion-based, but I think (1) and the original question is not. (This comment is not in response to Bryan Krause) – BCLC 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
  • @BryanKrause This question is not really about the GRE. This question concerns topics which are in the GRE. Using the term "GRE" therefore is shorthand the topics. Therefore, the question can be rephrased "For pure math PhD programs, what are essential topics after (insert the topics covered in the GRE without referencing the GRE)?" Thank you! Your comment has actually led me to think my question is off-topic here and should be asked on math instead. Should it? – BCLC Dec 12 '18 at 21:16
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    The subject GRE is a standardized test that can be optionally used by some institutions to guide their decisions about admission for graduate school. It isn't marketed or intended to test whether people "know math" or not, and I think most university math departments would prefer to set their own guidelines for their students rather than rely on the ETS. This question, and your last one, both seem to be arguing that the Math GRE does not contain the assessments of mathematics that you find relevant. And then your question seems to be "is there a right answer and if so am I right or are they?" – Bryan Krause Dec 12 '18 at 21:17
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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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    @AlexanderWoo Thank you! 1. When you say "US graduate programs", do you include or exclude master programs? That's an important distinction in a later edit in (2). 2. When you say math, do you mean pure math? – BCLC 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
  • @BryanKrause By graduate school do you mean PhD only or both PhD and master? I think you mean both PhD and master and if so, I think references to the GRE end here. Update: I think AlexanderWoo has answered (2), so I think we can end references to the GRE here. – BCLC Dec 12 '18 at 21:19
  • @AlexanderWoo I think you have answered (2). Thank you! – BCLC Dec 12 '18 at 21:19
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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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    @AlexanderWoo More accurately, the Subject GRE is a test of unpreparedness for both grad school and TAing calculus, in the sense that high scores don't mean all that much but low scores do. – Elizabeth Henning Dec 13 '18 at 19:59
  • @BryanKrause I edited the question. – BCLC Dec 16 '18 at 11:10
  • @ElizabethHenning I edited the question. By the way, in the previous edit, the question isn't exactly about the GRE. GRE is just a shortcut for "elementary complex analysis, elementary abstract algebra, elementary analysis and elementary topology" – BCLC Dec 16 '18 at 11:11
  • To answer your question, you really need to check with the US programs you're considering. Many, if not most, will describe their preparation requirements on their website. Nearly all will require the General and Subject GRE. – Elizabeth Henning Dec 16 '18 at 19:03
  • @ElizabethHenning I know, but what more than the subject GRE topics are required is my question? – BCLC Dec 17 '18 at 16:15
  • It varies. You need to check with the program. – Elizabeth Henning Dec 17 '18 at 16:23
  • @ElizabethHenning I didn't ask for every. I asked for the average. Same response? – BCLC Dec 17 '18 at 16:24
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    If you're fact-checking this professor, what he said is kind of stupid. If you're trying to assess your preparation for whatever an "average" program is, you need to check with them. – Elizabeth Henning Dec 17 '18 at 16:41
  • @ElizabethHenning Oh, thank you hahaha. For the "kind of stupid", I made a mistake. I should have said topology or geometry. I edited the post. Now it's not kind of stupid? And what if I define "average" in terms of uniform probability? List all the universities offering pure math PhD in the US. Pick one of them uniformly randomly. – BCLC Dec 19 '18 at 12:37
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    I honestly don't understand what you're after here. All programs will say they want an undergraduate major in math or a related field. But there are a few hundred pure math PhD programs in the US and their expectations are varied enough that even schools at roughly the same "ranking" will have different expectations. A mid-ranked school will probably expect basic abstract algebra and calculus with proofs. But it's ridiculous to make generalizations because each program decides what they expect and they often post this on their website. – Elizabeth Henning Dec 19 '18 at 16:57
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    They might also post their written qualifying exams, which give some indication of what they want you to know early on. – Elizabeth Henning Dec 19 '18 at 16:58
  • @ElizabethHenning The qualifying exams are for second year PhD students and therefore cover topics in first year PhD? My question is about required background of PhD applicants. My intention is to be properly informed, so I can make a decision about whether I should continue studying for Country A or apply to US style PhDs. If Country A is like the US, then I'll pick Country A because I live in Country A, and there's nothing to think about. If Country is unlike the US, then I'll think about whether or not I apply to US or US style pure math PhDs – BCLC Dec 20 '18 at 04:43
  • @ElizabethHenning Of course, we are talking about percentages or probabilities. If more than half of US pure math PhDs require only the topics covered in the GRE, then I have a decision to make. If we change more than half to less than 10%, then I don't have a decision to make. About variation, I shouldn't allow for many choices. Let us be binary: the university requires either only the GRE topics (1) or requires more than that (0). Maybe, I was not so clear about that. In this binary choice, is the average closer to 1 or to 0, in your best estimate? – BCLC Dec 20 '18 at 04:47
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    Yes written exams are taken early in the PhD program, but since a main goal of the admissions process is to identify students who will pass them, they are an indicator of what preparation is required. It is very likely true that more than half require no more than the GRE topics, if that is useful information. – Elizabeth Henning Dec 20 '18 at 04:52
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    I tried to clean up the text; it had drifted a lot and was becoming a small novel. Feel free to edit my changes if I botched anything, but I strongly suggest you try to keep it concise; all the details about "University A" made this straightforward question very confusing. – cag51 Dec 21 '18 at 05:16
  • @cag51 thanks! i edited further just now – BCLC Mar 27 '21 at 07:38
  • @ElizabethHenning thanks. i edited question more. i removed references to GRE. I included mention of john hopkins university and those US vs European questions i had asked years prior to OP. i can't believe i didn't include the US vs European thingy. i think my question just seems pretty confusing otherwise. anyway, please help further. – BCLC Mar 27 '21 at 07:39

2 Answers2

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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.)

Alexander Woo
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  • Alexander Woo, do you disagree with Buffy? Thank you! – BCLC Dec 12 '18 at 22:43
  • I edited my question. Is your answer mostly the same? Also, how about non-ivy leagues? – BCLC 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
  • Bu the standards of Professor A is just about me specifically. Does that mean you think I know enough pure math to apply for a pure math PhD in your university? – BCLC Dec 19 '18 at 12:28
  • What if I define "average" in terms of uniform probability? List all the universities offering pure math PhD in the US. Pick one of them uniformly randomly. – BCLC Dec 19 '18 at 12:29
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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
  • Alexander Woo: Comment 1: okay I mean average as in most. How about now? 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? COmment 2: Thank you! – BCLC Dec 30 '18 at 12:27
  • The only thing more than half (measured by pure count of programs, not weighted by size, which varies a LOT) of US PhD programs in pure maths require of almost every admitted student is demonstrated ability to write a complete and correct proof of a non-trivial fact. Almost every program will expect more than this, but the expectations are likely so varied that fewer than half the programs require any specific topic (and some programs will simply expect all of their students to have more mathematical background without specifically expecting any specific topics). – Alexander Woo Jan 03 '19 at 07:45
  • Alexander Woo, i edited question more. i removed references to GRE. I included mention of john hopkins university and those US vs European questions i had asked years prior to OP. i can't believe i didn't include the US vs European thingy. i think my question just seems pretty confusing otherwise. anyway, please help further. – BCLC Mar 27 '21 at 07:40
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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.

Buffy
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  • Hello Buffy. Thank you for your wisdom and empathy! I had edited and further edited to hopefully clarify. Some edits include my assumption that the GRE is for PhD only and not for both master and PhD and the phrase "first courses". Does your answer change? (The professor I spoke to 12 hours ago suggested some textbooks.. My question is supposed to answer my real question on whether or not the average undergraduate or the average pure math master graduate would know about these textbooks.) – BCLC Dec 12 '18 at 21:04
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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
  • On the GRE (my first edit): But ETS indeed expects some students to enroll in master (which is why its topics are what they are)? If ETS expects only PhD, then this sub-question might be opinion-based, but I think they should include algebraic topology instead of a lot of calculus? On the first courses (my second edit): Following up please! Every pure math PhD program would expect students to know, for example, algebraic topology right? I think this question is different from "Does every bachelor or master program in pure math cover algebraic topology?" – BCLC Dec 12 '18 at 21:12
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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
  • Hello Buffy, I already suspected that not all bachelor programs in pure math will include algebraic topology (I am surprised about general topology), but my question is if pure math PhD programs in the US will expect its applicants to know algebraic topology. Probably your answer is no because US PhDs expect only bachelor, but please confirm if I guessed correctly. – BCLC Dec 12 '18 at 21:24
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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
  • Thank you, Buffy! I just realized my question might be wrong. New question: Is algebraic topology expected of the average pure math US PhD program under the supervision of a professor researching in topology or geometry? I hope I included all the relevant adjectives, and I hope this may be the last follow up. – BCLC Dec 12 '18 at 22:42
  • Buffy to clarify, your answer to "After the basics of topology, abstract algebra, linear algebra, analysis and number theory, what are essential topics for any pure math PhD program?" for the average US pure math PhD program is "None" ? – BCLC Dec 12 '18 at 22:47
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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
  • @DaveLRenfro Thank you. It is not me. Country A seems to be leaning more towards European than US based on my interviews with professors even though the university websites look like they should be kind of US. One was surprised I couldn't describe what a manifold is, though I gave the correct definition of both manifolds and fundamental groups (I was unfortunately NOT asked to the describe a fundamental group). – BCLC Dec 16 '18 at 10:36
  • @DaveLRenfro Gee, if manifolds are so important, you'd think the GRE would include it, so we wouldn't waste time. He said first year PhD students in geometry would know what a manifold is, and this applies to all "reasonable" universities on earth. Would this be strange if this was the average US math PhD program but completely normal for the average European math PhD program? – BCLC Dec 16 '18 at 10:37
  • Buffy, I edited my question. Is your answer mostly the same? – BCLC Dec 16 '18 at 11:09
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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
  • @SolarMike How else can I get my question reopened, or do you mean I should instead ask a new question? That would risk a duplicate. So either way I'm screwed? – BCLC Dec 16 '18 at 11:33
  • Buffy and @SolarMike , i edited question more. i removed references to GRE. I included mention of john hopkins university and those US vs European questions i had asked years prior to OP. i can't believe i didn't include the US vs European thingy. i think my question just seems pretty confusing otherwise. anyway, please help further. – BCLC Mar 27 '21 at 07:40