Let S be a smooth projective surface over a field k. If S contains sufficiently many curves of nonpositive self-intersection, we show that that there are two such curves C_1, C_2 and positive integers a, b so that aC_1 + bC_2 has positive self-intersection. I will then describe some applications of this result to the geometry and topology of complex projective surfaces uniformized by the unit ball or the product of two Poincare disks. This is joint with Ted Chinburg.

I will discuss a continuous relaxation of the Cheeger cut problem on a weighted graph, and show how the relaxation is actually equivalent to the original problem. Then I will introduce an algorithm which experimentally is very efficient at approximating the solution to this problem on some clustering benchmarks. I will also give a heuristic variant of the algorithm which is faster but often gives just as accurate clustering results. This is joint work with Xavier Bresson, inspired by recent papers of Buhler and Hein, and Goldstein and Osher, and by an older paper of Strang.