I graduated from Caltech as a Math major. I guess I should have something to show for that...
Senior Paper
Caltech developed a "core writing requirement" between the time that I left and returned to get my degree, much to the chagrin of the students, who aren't so much into the writing. I'm not against writing, not that I wouldn't have minded more free time...
Regardless, I wrote my research paper on Constructive Mathematics. I was first introduced to the concept in a Philosophy of Mathematics course at UMaine. Basically there exist mathematicians that do not admit proof by contradiction. This is a bit terrifying, seeing as the business of proving things isn't exactly easy with proof by contradiciton. My paper gives a superficial treatment of the topic while building from the Natural numbers to the Intermediate Value Theorem.
The paper: A Brief Tour of Constructive Mathematics
My sources, which I follow very closely:
Varieties of Constructive Mathematics
Foundations of Constructive Analysis
Constructive Analysis