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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?

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closed as too broad by Matthew Towers, user299912, Juniven, projectilemotion, GNUSupporter 8964民主女神 地下教會 Feb 28 '17 at 14:40

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ @PeteL.Clark: I would really appreciate it if you could share your experience/knowledge/thoughts regarding the subject. $\endgroup$ – user400261 Feb 18 '17 at 16:01
  • $\begingroup$ I think asking this question at academia.stackexchange might be better $\endgroup$ – k99731 Feb 19 '17 at 5:05
  • $\begingroup$ I think you are right, I asked the same question there (academia.stackexchange.com/questions/85300/…). $\endgroup$ – user400261 Feb 19 '17 at 9:54
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I have had a look at the rankings. While I think it's true that math programs at the top of the list are stronger than those at the bottom of the list, I think it is silly to say that number $n$ is substantially better than number $n+10$ in any obvious sense.

Some departments are strong in some subfields and some in others, and in a ways that can change every few years.

The direct answer to your first question is that courses will tend to be more demanding and students will tend to be better at higher-ranked departments. But if 'level' of instruction means quality of teaching, that is not necessarily the case.

However, those may not be the issues that are most relevant to you. If you can get into one of the higher-ranked departments, you are probably used to being at the top of most of your classes. And all of your fellow students in the graduate courses can probably say the same. So you shouldn't necessarily expect to be at the top of your graduate classes, no matter how well-prepared you think you are. The competition may be an order of magnitude greater than you are used to. Your willingness to work really hard from the start and your motivation to keep that up for several years will have a lot to do with your success.

With regard to your second question: In general, both the size and the quality of the department will affect the number and variety of new ideas you are exposed to. It is certainly possible that you will find an interesting and rewarding thesis topic at a small, lower rated department. But the chances of starting a successful research career depend on finding one topic that fits your personal interests and capabilities. That will depend to some extent on the size and quality of the faculty, but it will also depend on your own interests and motivation, and also (to an extent) on luck.

One important issue you did not raise is how well the graduate department matches your personal goals.

  • If you are mainly interested in an academic research career, then a degree and with an excellent thesis from a top-rated department will be a real advantage in a very competitive job market.

  • If you are interested in an academic teaching career, then the quality of instruction to which you are exposed and your own perceptiveness about the ingredients in quality instruction may be the greatest predictors of your success.

  • If you are interested in doing applied research in industry or government agencies, then you should try to find a department where applied-oriented research and consulting are valued, and with a track record of graduates who have gotten jobs of the kind you want for yourself.

When deciding to which departments you will apply, you might use rankings like the ones in your link as some indication about how competitive the acceptance process is likely to be. However, you should do your own research on the departments. Do not rely on a program website to tell you everything you need to know. For example, you should consider what papers various faculty members have actually published in recent years above listed 'fields of interest'. You should also try to visit any campus that offers you admission. Does it seem to be a place where students are productive, motivated, mentored, and reasonably happy?

For several years, I had a government job that required me to visit many graduate mathematics departments. Some had much better records producing successful and productive graduates than others. Although I was not fond of the travel, it seemed to me that this kind record was highly correlated with aspects of the 'atmosphere' of a department that were not observable except by a personal visit.

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