This paper showed two century-old classification results, each of very undergrad-comprehensible things, were the same; pretty amazing!

http://arxiv.org/abs/1308.0751

"Sums of squares and varieties of minimal degree" by Grigoriy Blekherman, Greg Smith, and Mauricio Velasco

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of minimal degree.