Hint, quiet, don't let this out, a Ph.D. is not really a knowledge degree but a research degree. From that degree on, in blunt terms, everyone knows that no one can carry the whole library around between their ears and so no longer much cares what you know but cares what you can create!

There can be and are some exceptions, but overwhelmingly successful research in math requires the background of a Ph.D. for (1) finding a suitable problem and (2) having the knowledge to attack it. And it helps to be in a relatively good school so that will get relatively good versions of (1) and (2).

But with everything in good shape, apparently there is one more challenge -- being successful in the actual research. For a hint at this challenge, buried in D. Knuth's The TeXBook is:

> The traditional way is to put
off all creative aspects until
the last part of graduate
school. For seventeen or more
years, a student is taught
examsmanship, then suddenly
after passing enough exams in
graduate school he's told to do
something original.

That is, the research is work that is suddenly different, maybe for some people quite different and challenging, than all the academic work before. E.g., there are cases where a student made A's and was the darling of all the teachers from kindergarten through college but in all that time never encountered anything like having new ideas. Bad such cases can lead to stress, loss of self-esteem, crippled ability to work, more stress, burn out, clinical depression, and ... suicide. No joke.

For me, part of what helps in research is some qualified respect for some of the existing material. So, I look at what is there as needing improvement and try to do that. If look at the existing material as some nearly perfect construction, then maybe won't feel confident should or could improve on it!

One thing rarely taught in math is the importance of intuition: It is needed to do well at guessing, guess a suitable problem, broad outlines of a solution, attack, tools, etc. Good guessing is important since that's most of what there is to do, and good intuition helps with good guessing. Sure, when the results are obtained and in clean form with polished proofs, there can be little or no view of the sources, the intuition.

There can be some question about how good some Ph.D. research is: The professors don't want to grant Ph.D. degrees for poor research but don't really know how to ensure good work, indeed, for either the students or sometimes themselves. So one standard that can remove some possibly painful ambiguity is that the Ph.D. research should be "an original contribution to knowledge worthy of publication" with the usual standards for publication being "new, correct, and significant". If a student does some research and the professors question if it is publishable, then the student can settle the issue in an objective way -- try to publish the work.

E.g., computer science is concerned with computational time complexity, i.e., good algorithms where good means running time that grows no faster than some polynomial in the size of input data for the problem (rough statement -- more details in the famous

Michael R. Garey and
David S. Johnson,
Computers and Intractability:
A Guide to the Theory of NP-Completeness,
ISBN 0-7167-1045-5,
W. H. Freeman,
San Francisco,
1979.

and more recent sources).

IIRC that polynomial criterion came from J. Edmonds. More IIRC, he left his Ph.D. program early and did and published some of his work on networks. Eventually a committee of his former professors came to him and said that should he stack his publications and put a staple in one corner, that stack would be accepted as his Ph.D. dissertation and he would get his Ph.D.