In 1973 Serre proved that any action of the group SL(3,Z) on a tree must have a global fixed point. He deduced from this a number of beautiful results about splittings of groups and integrality of representations. In this talk I will describe a higher dimensional generalization of Serre's theorem, and will explain its implications in representation theory and combinatorial group theory. The main idea is a connection between the combinatorics of generators for certain matrix groups and the combinatorics of Helly's 1913 theorem on convex sets. This talk should be understandable to first year graduate students.