Edit: Right so I forgot this really big thing. My question is actually under this framework: Why are US PhDs different from European PhDs?
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?
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.)
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.)
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.)
You can see the previous revisions for more details.
Gonna copy some comments into the post:
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.)
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
(...) 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