I am an undergraduate mathematics student at some university in Turkey. I am doing well in math classes. Actually, I did not study very hard in my freshmen year and I got not so high grades from Calculus classes but then in the following years I studied well and got high grades from Abstract Algebra, Real Analysis and Topology classes. But because of my university's policy, I just took 20 math classes until my senior year. Here are my questions:
1) When I compare myself with a student in USA, I feel very inadequate. What should I do? How can I learn many more subjects?
2) Do I have any chance of acceptance to a PhD programme in USA? (For example, UC Berkeley) How can I increase my chance?
3) Finally, in the long run, do I have any chance of making a good contribution to mathematics? Because sometimes when I compare myself with others, I really feel inadequate.
Best Answer
Austin's and Matt's comments above are spot-on. I will just expand a bit on them here:
Regarding 1), as indicated in this thread, at the vast majority of US schools it would actually be impossible for a student to take 20 math classes by their senior year, as there are so-called "distribution requirements" that must be satisfied. Even the most dedicated, top US math students will have probably have taken less than 20, so don't feel pressured to attempt more because you feel you have to catch up. I should caution: make sure you are learning the material well! It is far better to take fewer math classes and really absorb what you're learning, than take a whole bunch, if the latter means that you are going to miss out on key aspects of the math. Good grades do not imply full comprehension (and vice versa, unfortunately).
If you still want to do more math anyway, I recommend simply reading on your time! It is a fun and free way to teach yourself some math that perhaps your university doesn't offer a class on, or review something you didn't get fully the first time around. You might also be able to audit a class you think would be interesting, but that you don't think you'd have time to follow fully.
Regarding 3), the frontier of mathematics is so vast (and always growing) that there is never any danger of the best people somehow "taking" all of the research. Perhaps you might not win the Fields Medal, but you will be able to make many original contributions to mathematics. It's best to focus on the enjoyment you personally derive from doing math; don't worry about the fact that there are X number of people "better" (whatever that means) than you. And you should take solace in the fact that mathematics is an extremely collaborative field - people are more than willing to share their knowledge with the community, as this site evidences quite frequently. There is no denigration of people for knowing less; we are all in it together.
Also, take a look at Terry Tao's blog post on this very issue. (Actually all of his blog posts are very positive and informative, I recommend reading them.)
There are also relevant posts on this site; the main one I can find at the moment is here.
Regarding 2), I'd like to know the answer too, as I'm also currently applying to grad schools :) But given your description, my (unqualified) opinion is that you have basically done everything for a grad school application as well as you could have done (your English looks excellent, and I assume you have gotten superb letters from your professors!) There may be some difficulties at US universities due to your status as an international student; I'm just guessing, but some places might have a quota. But even if that is the case, there's nothing to be done about it.
Again, this is my cursory, uninformed judgment; there are many professors here who have served on admissions committees at their schools, and you should not hesitate to listen to them over me.
As this is a community wiki post, I will shamelessly include Gerry's advice above, that you apply to plenty of so-called "safety schools" (i.e. schools that, while perhaps not as high a caliber you would like, you feel quite confident you'd get into).