I am unaware if this question has been asked anywhere on StackExchange before. However, I would like to know what is the major difference between a top 6 and a top 10 graduate school in Mathematics. Forex: Top 6 would include Princeton, Harvard etc. Top 10 would have a few more like Michigan or Columbia etc. I know that it depends a lot upon quality of theses and advisor. But can a graduate from Michigan be better than someone from Princeton? Looking at the faculty of top schools this rarely happens. Usually, everyone at Princeton received their doctorates from Princeton itself or maybe Harvard. Harvard/MIT also have the same situation. So is it really difficult for a Michigan(or any top 10 school) graduate to get a tenure-track job at a very prestigious school like Princeton ?.
I don't think you have gone very deep into rankings. Certainly not deep enough to suggest any relationship meaningful for individuals. I've never heard anyone scoff at a PhD from University of Michigan. You are suggesting that the top graduate from some place is invariably less skilled than the worst graduate from some other essentially similar place. That doesn't happen in the real world.
But if your life goal is to be a tenured professor at one of the top six schools in the US, prepare for disappointment, no matter where you study or how good you are. The problem is that the total number of open positions at such schools in any given year is minuscule and the competition for those few positions is huge.
But you should also think about the following in choosing a graduate school, if that is why you ask. Many top (and other) schools don't like to hire their own graduates. It certainly isn't that they doubt the quality, of course. But they like to bring in people who may have fresh ideas, not having studied with the current faculty and been exposed for many years with their ideas primarily. This has less effect now than it once did as it is easier to collaborate across distance and even internationally because of the internet and cheaper travel.
But some schools do permit hiring their own, and many students prefer to stay around a place where they are comfortable, so such places can collect a faculty made up of their own graduates. But, even in these, I'll guess that a lot of those "from here" people went out into the bigger world and made their reputation there before returning "home."
On the other hand, there are some incredibly fine mathematicians at quite small and out of the rankings places who simply want to live and work in a "home" environment of a different kind. Princeton isn't, actually, the finest place in the US to actually live. Too many cars, too many deer ticks.
But there is more to life than the quality of the institution you attend or the institution you work for. Lots more.
But can a graduate from Michigan be better than someone from Princeton?
Yes, of course. Consider how implausible the alternative is - that the sorting process as people applying to grad schools could actually succeed at sorting grad students so perfectly that the top 6 schools catch every top tier student and exactly those students.
Looking at the faculty of top schools this rarely happens. Usually, everyone at Princeton received their doctorates from Princeton itself or maybe Harvard. Harvard/MIT also have the same situation.
This addresses a very different question. Only a small fraction of Princeton/Harvard/etc math PhDs go on to become professors at Princeton/Harvard/etc.
It's quite possible that the best students from Princeton/Harvard/etc are slightly better than the best students from Michigan/etc, and that the faculty at Princeton/Harvard/etc are selected mostly from that very top group.
But the range of students at the very top schools certainly has a lot of overlap with the range from the schools a small step down.
Also, even the most cursory investigation of the actual faculty lists at Princeton, Harvard, and MIT, would reveal that the premise is false - all three schools do, in fact, have faculty whose PhDs come from other schools, both schools that might be in your "top 10 but not top 6" and schools that aren't in the top 10.
So is it really difficult for a Michigan(or any top 10 school) graduate to get a tenure-track job at a very prestigious school like Princeton ?.
Yes, it is really difficult for a Michigan graduate to get a tenure-track job at a school like Princeton. It is also really difficult for a Princeton graduate to get a tenure-track job at a school like Princeton