How much mathematics should one know before starting a Ph.D. program in pure mathematics? For example what topics one must understand well to pursue a Ph.D. in US University in Number Theory (preferably to work on Diophantine equations)?
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3$\begingroup$ More is better. $\endgroup$– user479223Commented Jun 20, 2023 at 20:28
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2$\begingroup$ You can look at syllabus for the qualifying exams of some schools you are interested in to gauge this. Here’s the website for Rutgers’s math qualifying exams: math.rutgers.edu/academics/graduate-program/… on this website is a pdf link to the topics covered. $\endgroup$– Sidharth GhoshalCommented Jun 20, 2023 at 20:32
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2$\begingroup$ Keep in mind you get to take classes before the exam. So you’re goal isn’t to master all the material BEFORE you show up to the PhD but to know enough so that within 1-2 years of starting your PhD you can master that material $\endgroup$– Sidharth GhoshalCommented Jun 20, 2023 at 20:35
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American students are generally not ready to start PhD because of mediocre undergraduate education. This is not really a complain about the US education (which I think is great), but because most of the US math majors are not interested in math and this is why the education is mediocre.
However, if you are a good, you are determined, and you study at a good PhD granting university, you can always find a way to study at high level. In the US there is no "no" and you can always negotiate replacing lower level courses with the graduate ones. However, vey few do.
You will be good for any graduate school if you feel comfortable with solving the problems from the book Berkeley Problems in Mathematics
If you think this is beyond your level you either went to a bad school or did not use opportunities the school provided.
answered Jun 21, 2023 at 1:31