I freeking love analysis and set theory. When I was a wee-lad taking algebra for the first time, I was completely appalled that a function was capable of taking a segment of the real line and mapping it to a curve that was longer than the original segment. Take, for instance, f(x) = 4/3*x. When x is between 0 and 3, the arc-length of the function is 5. How does it take something of length 3, and just by mapping points linearly, turn it into something of length 5 without ripping or taring anything?! Impossible!
Needless to say, when I finally got into an analysis course I was in heaven. And then Cantor's Diagonalization proof blew my mind.
The Banach-Tarski paradox is vaguely related, although even more absurd when you first hear about it.
Another oddity that freaks people out at parties is Gabriel's Horn, an object with finite volume but infinite surface area. Imagine: you can fill it full of paint, but never have enough paint to cover the surface! What's nice about this is it only takes knowledge of basic calculus to figure out, and if you're clever you can find the higher-dimensional variants as well.
A cool fact I proved accidentally while taking a number theory class is that the radius of the circle inscribed by any Pathagorean triangle is always an integer.
Actually, number theory might be my favorite subject of all within mathematics. I took my first class in it after about a year of grueling, highly-technical mathematics and physics courses. I described the sensation of being introduced to the subject for the first time to my friend (who's an astronomy major) to be akin to studying astrophysics for an entire year, deep within a library, buried in a dizzying pile of books, and then walking outside for the first time and seeing the stars again. That sort of simple beauty is greatly enhanced once you've been exposed to how enormously complicated life is.
My other love within math is dynamics and probability, which is probably what I'd go into if I went to graduate school. There's immensely interesting overlaps in the theory of zeta functions, spectral theory, ergodic theory, and dynamical systems, which all sort of tie together my interests in number theory, dynamics, and probability; although I admittedly am mostly ignorant of what they are, exactly.
I haven't taken abstract algebra or complex analysis yet, but my advisor says I'll love them. I've read a great deal about both topics, but I've been too busy lately to self-educate myself much. How have you guys liked those courses?
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