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Just out of curiosity, is there a topic where even an excellent graduate student wouldn't be able to master all the necessary background within 3 years (I say 3 because in many European countries a PhD is meant to last three years beyond the 5-year masters, with perhaps a fourth year to finish writing-up) ?

For example, I have heard that mastering EGA or Lurie's books takes several years, but does that mean some topics are simply out of reach in those European countries, and require an american-style PhD that lasts 5 years or more ?

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    $\begingroup$ In every research area there are problems that don't require mastering the full theory to make progress on, and problems that do. This concern exists for the second but not the first. $\endgroup$ – Will Sawin Sep 5 at 13:41
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    $\begingroup$ The responses in this question may help: mathoverflow.net/questions/335347/… $\endgroup$ – Stanley Yao Xiao Sep 5 at 13:43
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    $\begingroup$ I don't know about every area. Very few problems in algebraic geometry require mastering EGA. I don't know about higher category theory. $\endgroup$ – Will Sawin Sep 5 at 13:52
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    $\begingroup$ An American-style PhD doesn't really help with this issue, because students in such programs typically come in with only a bachelor's degree and spend the first two years on basic coursework, similar to what a European student would do during their masters. So the amount of time spent on research is about the same for both. $\endgroup$ – Nate Eldredge Sep 5 at 20:16
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    $\begingroup$ @Archie: There is an awful lot of info about this over at academia.stackexchange.com, which would also be a better place to ask further questions about it. The short answer is that American PhD programs usually have special provisions for students who come in already having a masters - they may waive some or all of the coursework requirements, or let them take an exam instead, or something like that. So such students may well be able to finish in 3-4 years, but are not necessarily required to. Every institution makes its own rules. $\endgroup$ – Nate Eldredge Sep 6 at 16:36

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