<p>On an unrelated note, everytime I hear or see the word “maths” my blood curdles and a little bit of my soul dies.</p>
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Of course they are probably all right around the same intelligence level; it would be absurd to try to distinguish between the student bodies as a whole in terms of intelligence. But I’m saying that the hardest class in MIT and the hardest class in Harvard might be more challenging than the hardest class at Princeton or the hardest class at Caltech, as the former two attract the majority of USAMO winners (basically, the top 12 math students in the country). And I would think that most, if not all USAMO winners, are highly attracted by what they perceive as the most rigorous courses available in Harvard or MIT. With that said, Caltech does attract a few USAMO winners, as does Princeton, so it’s not like I’m suggesting Harvard’s math 55 or MIT’s uber math whatever class is miles ahead of the competition.</p>
<p>^ All I was saying is that, you can’t really tell which school offers that hardest math class and the general quality of their student bodies alone is hardly a gauge for knowing that.</p>
<p>^ Once again, I’ll reiterate that I am only observing the trends shown by maybe 5-6 students every year. These are not typical type A overachievers that compose the majority of every elite college’s student body, but rather future Fields Medal recipients and Nobel laureates. They are naturally inclined to attend the school where they feel that they are being challenged to the limit, and so when the majority of these students choose to attend Harvard or MIT, it attests to the idea that these students believe that Harvard or MIT has the hardest math course available. Using Princeton or Caltech doesn’t really serve well to my example, as both also attract a fair share of math geniuses, but compare these four schools to, for example, Yale, and you’ll notice that while all 5 schools have similar student bodies, Yale has a dearth of IMO-caliber students because these IMO students are not attracted by even Yale’s most difficult math courses.</p>
<p>thanks everyone for the responses. has anyone used axler’s linear algebra done right?</p>
<p>also, people can argue about whatever they want to, but my opening question was more about the different speed of programs and their starting points rather than who has the most difficult course.</p>
<p>I don’t think the number of USAMO or IMO winners should determine the strength of the student body each university attracts. In fact, you could be a perfectly brilliant, impressive mathematician w/o preparing for/participating in those contests. Those who are interested in math in the early age do generally take part in them, but that is not to say that only the winners are good at math.</p>
<p>I honestly dont think that it has too much to do with the course offerings either. They all have grad schools. I think what makes everyone cluster is they want to be around where they think the best peers are just as the most famous professors pool at certain institutions. Plenty of brilliant mathematicians did not go to harvard/mit or princeton/caltech for undergrad and went on to much more competitive graduate programs than the middle tier of the students at these schools who when entering as freshman were likely considered better than the non-HMPC student.</p>
<p>I was in a similar situation as the OP, and I am now at dartmouth. I would say the following: the difference between schools with great grad pure math programs (mit, berkeley, stanford, princeton, Chicago for analysis, Mich for topology) is huge. I really wouldn’t worry too much about accelerated math programs, since they’ll compose a small amount of the math you’ll take, and most likely you’ll start taking higher level math classes quite quickly. I entered, completed calculus, finished baby rudin in a quarter, and now am on measure theory with folland. However, I am not covering as much material with as much depth and emphasis on creative problem solving as I would be at these top schools. Chicago is great for analysis but I would focus on overall strong undergrad programs with strong grad schools so you can have the benefit of smart students in both your classes and around you. Although Brown is great for applied math, the difference between applied math and pure math is nontrivial, and I really wouldn’t recommend Brown or for that matter williams/dartmouth/mudd(lack of graduate program) for anyone who plans to attend pure math grad school, if they could get into berkeley or michigan or chicago. That being said, who knows whether you want to go to grad school or become a mathematician. Maybe you want to be a banker or work in finance. Then dartmouth is great. Maybe you want to do applied math, then brown is great. </p>
<p>All that being said, I’d say go to the school that you like most after visiting. Four years is a long time and a lot can change during then, so you might as well enjoy it.</p>