I am going into my sophomore year as an undergraduate and I would like to ask the more experienced folks a couple questions about learning math and related things. What are your experiences and advice concerning the following dilemmas?

Being limited to a rate of 4-5 courses per semester, I realize that I am certainly not going to be able to take all of the courses that I am interested in. I would like to get a build a broad and solid base of knowledge by studying all areas of math at least a bit, but this comes at a cost of being able to take the more advanced, deeper courses. My plan was to self-study measure theory/Banach spaces and topology this year so that I'll be able to immerse myself in the graduate-level courses, which I expect to be more challenging and interesting and rewarding. I was wondering if people had experiences/regrets/wisdom about whether or not this is a good idea? Do you think it's better to build up a broad foundation thoroughly or throw yourself beyond your comfort zone?

On a similar note, what is your advice concerning specialization versus developing a broad taste? In my very limited experience, I have enjoyed representation theory, algebraic number theory, and complex analysis a lot. But there are still so many areas that I've yet to sample: algebraic topology, differential geometry, more advanced real analysis, algebraic geometry, analytic number theory, combinatorics... What's a good balance between trying all the different fields of math and trying to quickly become an expert in one?

Stepping back a bit, let me pose this question for a broader context. I am rather interested in philosophy, psychology, computer science, physics, and economics in addition to mathematics. I would like to take courses in these subjects as well but I am worried that this will put me at a disadvantage should I choose to ultimately devote myself to math. To people who chose either path -- regrets? hindsight? And of course, to anybody -- opinions on this issue?

and taken courses inthe foundational undergraduate curriculum in analysis (baby rudin), algebra (Artin) and topology (munkres). Is this the case? In general, self-study is a dangerous pursuit as an undergrad viewed through the eyes of grad admissions. $\endgroup$ – Justin Curry Jun 24 '10 at 11:14