<p>So I am a rising senior who is interested in math. I have a 4.0 in math and science at my school, but it really is not challenging and am going to end up only having learned BC Calc, though we will learn infinite series and stokes theorem. I am on the fastest math track at my school, but it doesnt move as quickly as i can learn.</p>
<p>I have done really well on math contests and get combined scores of 200-220 on the AMC+AIME (qualified for the USAMO once) and generally impress people with my creativity to compensate with my lack of knowledge of more advanced math.</p>
<p>Because I am starting with a relatively low point in the amount of math I know, where are the best place to go that have accelerated math programs where you dont have to already know a ton to be on an accelerated track. it is not that I cannot move quickly through the material, it is that i just dont know a lot of material that many other people do.</p>
<p>btw, my SAT/ACT scores are 2380/35 and my SAT II scores are 800,800,800 and will not be applying for financial assistance.</p>
<p>Honors Analysis at the University of Chicago is on par with Harvard 55, so you might want to look into that. (However, unlike Harvard 55, Honors Analysis is invitation only, and only about 10% of people who actually want to take the course get in.) I don’t see why you wouldn’t be able to do Honors Analysis (or Harvard 55, for that matter) after just Calculus BC, as long as you have a knack for the abstract. If you haven’t seen some of the basic components of advanced math, just look them up during the summer. Take out Rudin’s Principles of Mathematical Analysis. It’s not particularly difficult, and maybe five high schools in the U.S. even offer such a class.</p>
<p>In Honors Analysis, you start with Lebesgue measure theory and work your way up through a rigorous development of integration and differentiation (trust me, this is stuff that you haven’t seen before), L^p spaces, and then you get into abstract measure theory, signed measures, Radon-Nikodym derivatives, topology, manifolds, differential forms, Hilbert spaces, self-adjoint/compact/closed operators, etc. I don’t think the methods of MVC/ODEs/LA are needed anywhere here, except for a familiarity with vector spaces. Most of the material taught in MVC/ODEs/LA classes are calculation-based anyway and aren’t particularly helpful in lending the student familiarity with abstract principles of thought. Most important would be linear algebra, but getting acquainted with linear algebra isn’t terribly difficult. Learn the axioms of a vector space, get familiar with linear operators (i.e., matrices), etc.</p>
<p>^ I respectfully disagree about Rudin PMA not being that difficult. I finished my analysis coursework with Folland/Rudin Real & Complex, but I think most serious students would find Baby Rudin very tough. It’s a masters level, advanced undergrad text. For an intro to analysis, I think Marsden’s Elementary text is very good. Also Munkres’ Topology helps a lot to build intuition. In addition, Stein and Stikarchi (sp?) is a fantastic treatment.</p>
<p>To the OP, I think Harvard and Princeton would probably not be the best places to be an undergrad math major. Like MIT, Harvard gets a disproportionate share of IMO level kids who are really just awesome- unless you’re in the Berkeley math circle, son of a professor, etc. it’s gonna be tough to match these students’ experience. Princeton (like Caltech) has a small department in terms of majors, like 10-15 each year, but it is really geared to elite students as well. Most bright kids at Princeton (who aren’t certifiable geniuses) defer to ORFE or econ instead. </p>
<p>With your background, I think the U of C, Columbia, Brown, Penn, NYU, Cornell, Yale, Williams, Dartmouth, Swarthmore, Amherst, Wesleyan, etc. would all be solid choices- great curriculum with the level of difficulty you would probably want (all have honors classes if you’re looking for added challenge). And if you live in California, it would be tough to pass up UCB or UCLA.</p>
<p>Math 55 is unlike any other class in the country- math 230 at Yale is nowhere near as ambitious, nor is it overrun by IMO kids (who curiously, don’t apply to Yale much). </p>
<p>That said, I think Math 55 would be an absolutely miserable experience for kids not at the internationally competitive level… check out Curt McMullen’s website, he has all the lecture notes for his term of math 55 a few years back.</p>
<p>Caltech has the most rigorous math classes at the undergraduate level, followed by Princeton. MIT’s and Harvard’s courses are lighter in comparison. </p>
<p>The Math tripos at Cambridge (UK) is in turn tougher than any undergraduate math degree in the US, except perhaps Caltech’s.</p>
<p>Ah, so much disinformation. Most is hearsay and no one has any facts. Very few people who say that Harvard 55 is the hardest class in the world even know what its content is. So I propose that we start assembling the websites for the so-called top rigorous classes and judge for ourselves. The ones who have the experience to judge, at least.</p>
<p>I did “Fast Track Calculus” at Rose-Hulman Institute of Technology - Calc I, II and III compressed into 5 weeks - it’s certainly a lot of work since you do nothing but math all day long for 5 weeks but I doubt it’s as conceptually difficult as some of these Ivy League freshman math courses.</p>
<p>Brown recently had a graduate who completed his master’s in math and applied math in 4 years (though we only gave him a concentration in math and applied math + one A.M. degree). Not only do we have very strong math and applied math departments, but you have full access to graduate course work and high level course work from day one, no problems, no questions, and can take as much math or applied math as you feel is necessary.</p>
<p>Plus, being a smaller school, you’ll have plenty of room to distinguish yourself and have professors jumping on you for research opportunities.</p>
<p>I’d say we’re a definite top choice for exceptionally gifted math students to have a great undergraduate education while pursuing very high level math for an undergrad.</p>
<p>if i wanted to teach myself multivariable calc and/or linear algebra, does anyone have any recommended textbooks that are good for do it yourself?</p>
<p>Why not just take advanced versions of that course work first semester? They can easily be taken simultaneously, and once you’re through linear algebra you can pretty much take any math course you want at most schools without too much sequence (except for the year of abstract algebra).</p>
<p>i guess i dont understand the suggestion. are you saying to take advanced versions of the course first semester in highschool or in college? my highschool does not offer linear algebra or multivariable so i was looking for some books where i could teach it to myself</p>
<p>First semester college, take the honors version of both. They’ll be a great introduction to how to do math in a college setting and prepare you for everything else. There’s no need to jump in at the graduate level, you’ll find there is plenty of math to do even if you take a semester to finish that sequence. In fact, at Brown, which again, has stellar math departments, the honors version of MV calculus is easily one of the most challenging courses in the curriculum.</p>
<p>I would have to agree with modestmelody over here. At Brown, if you want, you are completely free to take 5 math courses each semester every semester. Not that you’d want to, but you have the option of catching up and reaching the level you want as fast as you want. Also, 95% of courses are designed in such a way that they only require two basic courses as prereqs (multi.var. and linear algebra - which you can freely skip anyway). </p>
<p>And then, you can start doing research from day 1 with whomever you wish in the department. Whether you know what you would like to work on, or have no idea whatsoever. </p>
<p>And last but not least, taking a class with our prof. Banchoff over here I can guarantee will be the best experience of your undergrad career. What other world renowned profs like he do you think would spend at least one-two hours per week with each of his students (even in a class of 30), individually checking reading and discussing homework and exams, for as long as needed until the student understands how to do everything by himself? I am serious, almost all day long you can post a message on his very own forum-like system, and he’ll give feedback right away. Had an idea? Ask him, he’ll have comments. Want to hear a good story about when he met Nash? Ask him, he’ll be very excited to share. Is your completely original solution very good, but still lacking that 0.001% to make it a real proof? He’ll teach you how to handle it, and then tell all the students what an amazing solution you found.</p>
<p>So, be quick, apply fast, and you can catch him teaching differential geometry next year! An opportunity you shouldn’t miss whether majoring in math, physics, bio-chem or arts! I am not kidding! I mean, he was bff with dali</p>
<p>bruno123, yeah, the maths at Cambridge are oversubscribed and only the best of the best are admitted. it’s probably Cambridge’s strongest program.</p>
<p>I’d also include the ff unis as options:
Oxford
Warwick
Imperial</p>
<p>that if the OP is considering on studying in the UK.</p>
I do not know the curricula at any of these schools, but the fact that MIT and Harvard attract more top USAMO scorers than the other 2 attests to the idea that the reverse is true, although perhaps not as a whole but in terms of each school’s most rigorous class.</p>
<p>I think Harvard, MIT, Caltech and Princeton have pretty much identical student body in terms of intellectual capabilities. You can’t really say the students in Caltech are smarter than those in HPM and vice versa.</p>