1. Different subfields connect to each other in awesome ways. I'm doing graph theory research this semester with a group on campus. Most of my graph theory experience is in one subfield (edge coloring), and the group does work in another subfield (vertex labeling). Within my first couple of weeks with the group, we solved a vertex labeling problem by changing it to an edge coloring problem and solving that, which I thought was really cool.
2. It explains weird phenomena. This past Saturday night was the first ever Midnight Mathematicians at which the speaker was a professor. Midnight Math meets every other Saturday night (ish) at 11:59 pm, and we all dress classily, eat cheese, and listen to someone talk about math! This weekend, Professor John Geddes talked about work he's done on nonlinear dynamics of fluids, particularly related to blood flowing in microvessels. The blood sometimes does weird things, like change directions with no change in conditions, and the mathematics can actually explain this, which is so awesome. (I in no way mean to imply that math is only cool if it is useful. I've done and enjoyed too much model theory and logic to think that.)
3. And finally, it leads to exchanges like this:
(In Partial Differential Equations on Friday, talking about the wave equation on the half line. Aaron is the professor.)
Student: "That's why we can't make a semi-infinite guitar!"
Aaron: "That's why we wouldn't want to. The finiteness of the universe is why we can't."
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