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Initially I asked this question on math stackexchange but it seems academia stackexchange is more appropriate place for this.

Note: a question similar to that I'm going to ask was discussed here, but answers to the questions I'm going to ask were not covered there.

Basically, I'd like to understand how significant is the gap between lower- and higher-ranked graduate schools in mathematics. I'll be referring to the USN graduate schools ranking. Let me fix the notation straight away. By 'higher-ranked schools' (which I will also refer to as 'top schools') I will mean top 10 (or if you want top 20) and by 'lower-ranked schools' I will mean schools ranked 20-40 (all according to USN).

What I understood from the question I provided the link to above is that being amidst 'top students' at a higher ranked school is more beneficial than being amidst 'average students' at a lower-ranked school. Also, higher-ranked schools may be more diverse and provide more opportunities in meeting people from other top schools as well as in obtaining a job in the academia.

But I still have a couple of questions.

The first one is about the first two years of the study. Is, in general, the instruction level at lower-ranked schools worse than that in higher-ranked ones? Also, are graduate corses at 'top schools' more difficult to master and to pass? If so, does it imply that one needs a better preparation (i.e., a stronger mathematical background) to succeed in a top graduate program? Also, does it imply that the students enrolled in a top program will eventually have a better mathematical background that will make it easier for them to conduct research? If you have anything else to say about the coursework at top universities in comparison to that at lower-ranked universities, I would appreciate it.

Secondly, in the question I referred to above (or elsewhere), some people mentioned that exposure to new ideas in various branches of mathematics (which is one of the advantages of top programs) is a consequence of the size of the department and the 'quality' of faculty/post-docs/students. Whereas I do agree that students at top universities are more knowledgable and creative, I cannot see why the other assertions hold. Correct me if I am wrong but the the math department of say Stony Brook or Indiana (ranked 25 and 34, resp.) is not smaller than that of Chicago or Columbia (ranked 5 and 9, resp.). Furthermore, the vast majority of professors in all of the places mentioned are alumni of Harvard/Princeton/Berkeley/MIT/Chicago/Stanford (i.e., a top school in my terminology); post-docs also come from very prestigious places to all of the four mentioned universities. So what makes Chicago or Columbia 'better' than Stony Brook or Indiana? Just their name? (Of course Chicago may have more publications than Indiana but I don't think that graduate students feel it.)

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    better faculty = > better advice – Rüdiger Feb 20 '17 at 22:27
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    There's a lot to respond to here. Keep in mind that those graduates of the top schools are not teaching at the top universities, so perhaps they were poor performers at better schools? As a very general answer: the graduate experience is what you make of it. I had poor students when I taught at top schools and stellar students when I taught at not-top schools. If you want to get into academia these distinctions might matter a little but if you don't, then probably -- as you already seem to intuit -- not so much. – Dave Kanter Feb 27 '17 at 20:12
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    in my personal experience the rankings are useless. I am a mathematics PhD student at a university with some of the most accomplished topologists in the world, and yet my graduate program gets very low rankings. Connect with the right people and have a bit of luck in your research. – Forever Mozart Mar 23 '17 at 3:53
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Looking at the problem as someone who has been on a variety of hiring committees and grant panels, I would say that in general, having someone with a good reputation in the field writing a strong letter of support is the most important factor, together with the journals where you have published.

In view of this, the school doesn't really matter: if you do your PhD in a small school but with a supervisor with a solid reputation who produces a multi-page letter highlighting well your contributions to the field, you will usually fare better than if you come from a very well known school with very little endorsement from your supervisor. Of course, some people tend to get a little bit star struck, but the dynamics of a hiring (or funding) committee usually smooths things out. Actually, contrary to what Dave Kaye was saying, I would argue that the school matters less in academia, more if you want to venture out into the real world, where name recognition is more important and might help you land your first job.

So why favor a good school? Well, there will be more visiting speakers of a high caliber. There might be a wider variety of grad courses to choose from. Overall, probably a more enriching experience. The difficulty level will not be very different, if at all, once you have been admitted.

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I know I'm quite late to the party, but I disagree rather strongly with part of the other answer, namely:

  1. Yes, the level courses are taught at can vary quite a bit between first tier schools, second tier schools and third tier schools. (Also though the drop off is probably steeper going from second to third tier.) Consequently, it may be much harder for someone to succeed in the first couple years at a top school than at an "average" school, particularly if one does not have adequate preparation. This is not to say it's impossible---I've seen PhD students need to start with undergrad courses in a top 10 school and then successfully graduate. In fact, I think the success (PhD completion) rates tend to be higher higher at top schools, because of course they try to only accept the students who seem likely to succeed.

  2. Generally top schools have the professors doing the most impressive/popular (subjective, I know) research. It's not the size of the department or the number of publications that varies so much, but it's the quality of research. It's not just the name. That said, rankings themselves are rather imprecise---certainly departments are not linearly ordered by quality---and some schools are better in certain fields than others: maybe a second tier school is actually one of the top schools in your field of interest.

In summary, you will certainly have a different educational experience at Chicago than at Indiana. This does not mean you will have a better experience at Chicago---this is highly circumstantial/individual, or that you would get a better job coming out of Chicago, but the readily available resources will be different. Interactions with the leaders of your field is a very valuable thing.

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