I am currently a senior in high school, and I have been studying mathematics for about nine or ten years now in my personal time outside of school. I am not familiar with academia or in general higher education, but I do know that I want to continue studying mathematics or something in a related area.

I am having a lot of trouble pinning down which universities I should consider going to, I don't think I am even ready to start comparing, so despite the 'softness' and vague nature of this question I feel I don't really have any where else to start then by asking which universities should I be looking into for mathematics?

I have read many articles that rank schools according to some criteria, articles with titles like "Top Universities for Mathematics" or "Best Mathematics Programs", but I really want to get the opinions of people who study mathematics frequently. Any school suggestions would be greatly appreciated. I am trying to gather a list so don't hestiate.

I live in California and am willing to go anywhere on the planet, so location is not a problem for me.

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Are you sure that you want a large school? There are smaller schools with excellent math programs, like Pomona College and Harvey Mudd College, and contrary to what @James just said, it is not necessary to attend an undergraduate school with an active graduate program in order to be pushed. – Brian M. Scott Oct 7 '13 at 3:04
University of Michigan Ann Arbor. UCSD has very good (and friendly!) mathematicians, too. I got some great advice when I was applying to undergraduate school from a professor. Go to the school that suits your personality and learning style best for undergraduate, specialize when choosing grad school. It's very unlikely that you'll reach a level advanced enough to be beyond the teaching ability of your professors as an undergraduate, so you want to go to the school where you'll learn the most efficiently. – Callus Oct 7 '13 at 3:25
@J.W.Perry: Or better yet, combinatorics. But the really nice thing is that if you’re at any of the Clarement Colleges, you can take those Harvey Mudd courses if nothing comparable is available at your school! I was at Pomona but took all three of my topology courses and my logic course at HM. – Brian M. Scott Oct 7 '13 at 3:25
A major downside of the University of Virginia: it's in Virginia. – dfeuer Oct 7 '13 at 3:52
@dfeuer : Your comment is offensive to no more than roughly this much of the U.S. population. I am no longer a resident, but the region around UVA and the university itself is, in my opinion absolutely amazing and one of the nicest places in the country to live. I have attended graduate ceremonies at the same for immediate family. – J. W. Perry Oct 7 '13 at 4:10

Northwestern is a fairly small school with an excellent math faculty, who make a real effort to identify freshmen who are talented in mathematics and direct them into appropriately challenging courses. There is plenty of opportunities for math majors to get to know individual faculty members, do independent studies, write senior theses, take graduate courses, etc.

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Matt E: what about Chicago? – Alex Youcis Oct 10 '13 at 3:17
@AlexYoucis: Dear Alex, It was already on the OP's list, so I don't need to say anything about it! (Not surprisingly, I think it's a good program. There are lots of opportunities for advanced courses, as well as independent study with faculty.) Cheers, – Matt E Oct 10 '13 at 3:58
@MattE I didn't know if there was some preferential nod to your old institution ;) Thanks for clarifying! – Alex Youcis Oct 10 '13 at 4:01

Waterloo shows up high on Putnam lists, so you might want to consider it.

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Yeah look into the University of Waterloo. It's an excellent school for Mathematics. – user1804933 Oct 10 '13 at 1:37
I did my undergrad at Waterloo in the Math department, and I will be starting a Masters there as well. It is definitely a great school for Math/CS/Eng. and the like. – Nizbel99 Oct 11 '13 at 14:16

If you are interested in any university in the world, you should definitely consider the UK.

Cambridge has a very strong mathematics programme, developed from a long heritage. Oxford, Imperial College, Kings and Warwick are also very good in the UK.

A comparison website with an international bent is http://www.topuniversities.com/university-rankings/university-subject-rankings/2012/mathematics

Aside from all these "top ten" type of lists and people's opinions, you should find a university which resonates with you personally. You will get the most out of the course if it fits you, not if it is the best programme ever.

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The University of Manchester is pretty good to. – Vedran Šego Oct 15 '13 at 13:57
Note that Oxford and Cambridge require applications in by the $15$th of October. – Alyosha Oct 15 '13 at 16:22
Cambridge & Oxford also both have tests involved - for Oxford you do a test to decide if you get onto interview, and for Cambridge if you get an offer it is nearly always (with exception to people with sufficiently good Olympiad placements) conditional on the STEP papers. – Andrew D Oct 15 '13 at 16:41
I don't know about international undergraduate applications. From the UK, A-levels can be sufficient, but often a STEP examination is asked for. Here are a few links: study.cam.ac.uk/undergraduate/apply and study.cam.ac.uk/undergraduate/courses/maths – Chogg Oct 17 '13 at 11:49

University of Toronto is great too.

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I'm biased though since I go to U of T. – Pratyush Sarkar Oct 10 '13 at 2:37

To expand a bit on Chogg's answer ("find a university that resonates with you personally"), it's important to decide what you need/want from your education. The views below are largely orthogonal to the replies you've received so far (in particular, no specific schools are suggested), and are based on my experiences in the US and Canada.

Putting aside practicalities such as cost and geographic location, salient items to consider include:

• Your intented profession and expected final degree (e.g., bachelor's, doctorate).

• Your own mathematical abilities (which can be difficult to assess until you have basis for comparison).

• The type of academic environment in which you work best (e.g., competitive, supportive).

Generally speaking, a large research university offers courses in a wide range of subjects, at levels from introductory up to research level. The intellectual environment is rich; you can attend colloquia and seminars on topics directly related to (or far from) your studies, socialize and study with with bright students, and (depending on the surrounding area) encounter life experiences (culture, outdoor activities, urban--or rural--life in general) that broaden you as a person.

You'll have a percentage of classmates who are Much Smarter Than You, i.e., who pick up new ideas more quickly, who have seen much more mathematics, perhaps even some whose thought processes are "magical" in the sense of Mark Kac. (This is all the more true in graduate school.) The intellectual atmosphere is stimulating if you're inspired by genius, and often crushing if you're the smartest person you've ever met.

Courses are large, with correspondingly less individual attention from faculty; you'll learn much of your basic undergraduate material from teaching assistants. If you shine mathematically, you'll earn the attention of faculty, and will have no trouble getting individual attention that will take you wherever your abilities allow. If your mathematical abilities turn out merely to be "standard", you can still get an excellent education. However, if you don't perform well without personal attention, you can easily "sink" rather than "swim".

By contrast, a smaller school generally offers smaller courses (with more individualized attention from faculty), at the expense of generally limited course offerings and faculty expertise, a smaller student body (a narrower range of interests and abilities), and fewer colloquia and seminars. However, at a smaller school you're less likely to feel "lost" or "abandoned". If you're lucky enough to find a small school with a faculty member doing research close to your interests, that can counterbalance most of the general disadvantages of a small school.

One final thought: You may be able to get advantages of both types of school by enrolling full-time in a small school located near one or more large research universities. At the very least, you can sit in on courses and seminars as your schedule permits.

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Excellent explanation! I'd re-emphasize that if one has no idea how one'll react to encountering even-more-capable people, and/or seeing genuine bleeding-edge mathematics done by mature mathematicians who're a decade or more "grown up", going to a big university is more likely to make this clear than going to a small four-year school. It might be disheartening in some ways, but it is much more informative about the larger reality. That is, there is a definite hazard in being allowed to continue thinking that one is the smartest person on the planet, etc. Better to get over that conceit early. – paul garrett Oct 16 '13 at 0:02

CMU also shows up high on Putnam lists, it is a great place to study math unless you want to do research in geometry. :D

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you are welcome! and good luck! :D – user67133 Oct 10 '13 at 1:58

You've got to talk to recent alumni of the schools you are looking at who attended at the level you are going to (undergraduate) and the departments you are looking at. There is a substantial variability of student experience within each school, depending on the level, department, and year, in that order. I attended UCLA as a math+CS graduate student, and was quite happy; my niece went there 10 years later as an undergrad chemistry student, and hated it enough to transfer out.

One thing to consider is the dynamics of funding of public universities, or lack thereof. Due to budget cuts many public universities cut the maximal number of units you can take, presumably because you pay them proportionally to semesters attended and they have to employ the number of lecturers proportional to the number of units the students take each semester. This effectively turned the 4-year degrees into 5-year ones.

You should spend some time at prospective campuses during school year beyond the usual tours so that you'd get a feel what's it like there, perhaps audit a couple of lectures. In the universities that hire professors primarily for their research achievements rather than teaching abilities many undergrads find it impossible to understand the lecturers, either due to the foreign accents or due to ill-prepared lectures. This makes some of the schools that are among the best for graduate students and researchers ironically sub-par for undergraduates.

The roads for the best education differ greatly for different people. IMO the best students, the creme de la creme, should go directly to those schools you called "insanely prestigious", where they could get engaged in undergrad research much sooner. However, for the B+ students the best choice may be splitting the undergraduate education into the lower division half in the local community colleges and the upper division half (via transfer after AS degree) in a top 30 university that has a good program of the student's choice. The reasons: first, in the community colleges education is very cheap and the lecturers are as good as anywhere: one doesn't have to be a Fields Medal winner to teach 1st year calculus; second, by the time the student gets his AS degree he has a better understanding of what he wants for the better choice for the university and department to transfer to.

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If you are really interested in universities around the world Singapore is always considered among the best in the world. It's expensive and I don't have much personal experience there, but you could have a look.

If you would also consider countries where English is not the main language:

Continental Europe also offers a few very good universities that you might not have on the map because they don't rank first in the usual rankings. The ETH Zürich is definitely among the top universities (as an undergrad, I doubt it makes much difference going to Harvard, Oxford or ETH) and all students I know there have been quite happy. Also, the level is quite high from the start (problem: Switzerland has quite high living costs and the universities are discussing charging foreign students higher fees). The École Normale Superieure in Paris also crops up at these rankings and is among the elite programs of France. Germany of course, also has a few very good maths programs (most famous are perhaps Bonn, where you can also find the Max-Planck institute for mathematics and the two big Munich universities). Since the system is much less elitist than in the states, the top universities also have less money, which of course you will notice. Also, universities can't really choose their undergraduates (no SATs or similar tests; mostly for mathematics, an "Abitur" or something equivalent is all you need), hence you won't have the elite crop like in Oxbridge or the Ivy League. That, however, doesn't mean that the programs are bad (if you are not good enough, you'll drop out fast).

Generally speaking, though, undergraduate programs in continental Europe differ a lot from your undergraduate programs in the states (which is something you should inform yourself, if you really consider crossing the Atlantic). For example, you won't find "Calculus" courses, you'll directly begin with proof based real analysis and linear algebra. Calculus is already taught in school and will be refreshed during real analysis. Of course, these "analysis" lectures start out pretty basic, but they are already in the spirit of all that is to come. I guess it's a matter of taste.

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I'd like to add some info on France: Universities Paris-6 (aka UPMC), Paris-11 (aka Paris-Sud), Paris-13 (aka Paris-Nord) have good maths programs, too. If you can enter so-called Grandes Ecoles, you should consider Ecole Polytechnique, Ecole Normale Superieure, and Ecole de Mines. They are always in top-250 in rankings and they have really good programs in maths and other sciences. – TZakrevskiy Oct 15 '13 at 16:08

Since you are asking for a big-list, you might add Australian universities. But I doubt you want to travel to the other side of the world.

University of Melbourne, Australian National University, University of Sydney are the top ranked schools. Many good students from these universities are accepted into top US and UK graduate programs. Even their PhD graduates work in overseas Postdocs and other academic positions.

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Drake University is my Alma mater. Easy to double major in Computer Science since they share faculty. One of the best undergraduate Actuarial Science programs in the nation. Lots of good relations between Math and Physics faculty. Lots of students study abroad in programs like Budapest or do summer NSF REUs. Lots of local insurance companies to do actuarial summer internships with.

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have you heard of math 55 A and B at Harvard? Richard Stallman and Bill Gates did it and if you can, then you can move to grad level courses. I know you solved 400+ Euler problems

https://en.wikipedia.org/wiki/Math_55

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