If you like the fun math, go with abstract algebra for sure, number theory (though until you know abstract algebra and complex analysis I'm not sure what you can really do in number theory), and combinatorics and graph theory. Those are my suggestions!
In my opinion, topology or abstract algebra would be the most important, in part because a lot of other courses will rely on these. At the undergraduate level a Geometry course will most likely be a course in Euclidean Geometry. I took such a course as an undergrad and really enjoyed it, but unless you are specifically interested in this you should postpone taking it until you get core classes out of the way.
If you can take game theory over the summer definitely do so, its a really interesting and fun subject. You will likely encounter some elementary theorems from topology in game theory, most notably some incantation of a fixed point theorem, so this will help you a bit in the fall if you end up taking topology.
If you plan on going to graduate school for math, then take as much analysis (real and complex), algebra, and topology as you can.
I thought they were similar because, where I went for undergrad, abstract algebra was a pre-req for number theory, and *I thought* a lot of number theory results rely on algebra...
Thoughts on what I should take for my 2nd class this summer and then during the year? Was thinking something like:
Game Theory (summer)
Abstract Algebra
Probability Theory
Optimization
Complex Analysis
Not sure whether I should sub in topology or geometry for one of the above...thoughts?
Thanks for the help, Chezlaw.
Algebra is used a lot in number theory. However Algebra is used in many fields and it's incredibly useful (especially linear algebra, you can never know too much linear algebra).
Of Real Analysis II, Probability, and Game Theory, you should take a second semester of real analysis, and it's not close. Probability is second, and game theory is a distant, distant third.
Algebra & Number Theory are not a waste to take together, but you should focus on your core competencies and your GREs if you're thinking of grad school. There's plenty of time for number theory later, and it will be richer as your algebra and analysis backgrounds improve.
I would much rather see a guy coming in with 2 semesters of real analysis, 2 semesters of algebra, an intro to geometry, point-set topology, and complex analysis than any of the other stuff. The other classes are great as electives, but I think a solid grounding in the basics is so so important.
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