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I am an undergraduate Math student, i find the courses that I am in are too easy (calculus, linear algebra, Discrete math, logic).

I want to become a successful mathematician. I want to know what material i should study by myself? Should I just do all the extra problems in the book, or should I start reading about general problem solving, or maybe about a specific topic..

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I myself have learnt a lot by "hanging around" at math discussion boards (like this one), math blogs, and by discussing on the Internet. I guess reading an introductory book on group theory or something similarly abstract could be what you need. – Fredrik Meyer Jan 20 '11 at 2:46
You are an undergraduate student in what major? Computer Science or Mathematics? – Aryabhata Jan 20 '11 at 2:47
Take Grad classes! (or at least ask the instructor's permission to sit in on them) – jericson Jan 20 '11 at 4:16
@jericson: I disagree. I have seen the folly of (even very bright) undergrads who try to take graduate algebra / analysis / topology without first taking the undergrad variety. This just sets up future problems for no good reason. – Pete L. Clark Jan 20 '11 at 6:11
@Pete L. Clark: Thats fair.. To me it depends on a combination of one's mathematical background/exposure, the difficulty level of the undergrad/grad classes at the institution, and the class sizes/instructor flexibility. – jericson Jan 21 '11 at 2:19
up vote 14 down vote accepted

I find it telling that you do not list any intermediate level courses among those you have taken: especially, real analysis, topology or algebra. These are the courses which provide the theoretical foundation for all later study of mathematics (and are, as in Roy Smith's answer, arbitrarily challenging: one cannot be too good for any of them).

Have you not taken these courses just because you are in the first or second year of your undergraduate program, or for some other reason? Is it too late in the current term for you to take one of these courses? I think it might be worth a try to transfer into them.

As with many times when an undergraduate looks for help from the internet masses, I also wonder: are you being properly advised? Do you have a faculty member in the math department at your institution with whom you are discussing these issues? If not, you should find one right away. "I am not being properly challenged in my classes and would like to learn more interesting mathematics" is just about the best thing a faculty member can hear from a student. It is hard for me to imagine that you will not be received with open arms.

Added: I see from your profile that you are a first year undergraduate at a Canadian university. That actually explains a lot. Most North American undergraduates are not ready for the sort of theoretical mathematics I described above as first year students -- but some are. Accommodating both groups is a serious curricular challenge. At some universities there is a sort of "honors track" for those who are: e.g. both at UGA and at Chicago there is an honors calculus course taught out of Spivak, and at both Chicago and Harvard they have very challenging analysis courses for exceptionally strong first year students. But you have to be both well-prepared and well-informed in order to place into these courses. Moreover, I have taught at two Canadian universities, and my feeling is that they are a bit more sticklers for taking courses in a fixed sequence than American universities of comparable quality. One of my closest friends started an undergrad degree at UBC at around the age of 15. He did extremely well in his classes from the very beginning, but nevertheless took a lot of "cookie cutter" math courses (e.g. differential equations without any real analysis) that such an obviously prodigious student might at a top US university be well-advised to skip. Anyway, he got to the good stuff at the advanced age of 17 or so, and he is now a successful grown-up mathematician. So be aware that that's the local culture to a certain extent. But I stand by my previous advice: contact a faculty member and let them know what you're feeling. At the very least you should be able to find someone to talk to as you work through Spivak, or Little Rudin, or Artin, or whatever.

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+1. This is exactly the right advise for someone in the OP's position – jericson Jan 21 '11 at 2:33
Thank you very much, I will do as you advised. – Lux Jan 21 '11 at 4:39

In addition to "general problem solving" (I second PEV's recommendation of Polya's book), much along the lines of this previous question about what to do after Calculus, I would suggest to start with Number Theory and History of Math; you can learn a lot about both on your own. I would suggest sticking to "elementary" number theory to begin with on that side (that is, not "Algebraic Number Theory" and not "Analytic Number Theory").

If you are still thirsting for more on your own, you can probably do a fair amount with some basic point-set topology, some graph theory, or maybe even some advanced calculus/introductory analysis.

While old IMO problems or Putnam problems might be interesting, I would not necessarily go there myself; while there are a lot of really good mathematicians who excelled at either or both, there are also a lot who did not. Rather, I would perhaps suggest leafing through the Mathematics Magazine (probably available from your Department); it includes a problems section if you want to try your hand at that kind of thing, but will also have other articles, many of which will likely be accessible.

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Read harder books, like Courant's or Spivak's or Apostol's Calculus, and Van der Waerden or M. Artin on algebra and linear algebra. If those are too easy try Spivak's Calculus on manifolds or Fleming's Calculus of several variables, or Dieudonne's Foundations of modern analysis and Jacobson's Basic algebra I, or Chi Han Sah's Abstract Algebra, or Fulton's Algebraic curves. If those are too easy try Hatcher's or 'Spanier's algebraic topology and Lang's Algebra, and Arnol'd's Ordinary differential equations. Or Spivak's Differential geometry, or Rick Miranda's Algebraic Curves and Riemann surfaces, or Shafarevich's Basic algebraic geometry, or Siegel's lectures on Riemann surfaces, Gauss's Disquisitiones, ..........

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I recall that you said that Maurice Auslander told you to write 10 pages for every page you read or else you have learned nothing? – PEV Jan 20 '11 at 5:45

Here are some possible resources for problem solving:

This (MO post about how to go about studying undergraduate math) advice seems helpful.

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what is your rationale for recommending that the OP look at old IMO problems? Speaking personally, this is among the last things I would recommend: to me this looks like high school trickery, whereas the OP seems to be yearning for university level math. But perhaps this is just a reflection of my own distaste (and relative lack of ability) for this sort of thing. If you have actual experience that solving IMO problems helps bright undergraduates, I'd be interested to hear about it. – Pete L. Clark Jan 20 '11 at 6:09
@Pete L. Clark: For problem solving purposes, looking at IMO problems wouldn't hurt. I think it just sharpens the problem solving mind (i.e. your box of tools). But I agree that it's not "real math." For example, see this:… – PEV Jan 20 '11 at 6:34

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