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?
Advanced real analysis (the one with measure, Lebesgue integration, Radon-Nikodym, etc)
Partial differential equations
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.