Abstract: A basic problem in algebraic topology is to write down all homotopy classes of maps between spheres. This problem, simple to state, is impossible to solve -- we don't even have a working guess. However, we've gotten very good at using some very specialized bits of algebraic geometry (the theory of abelian varieties and $p$-divisible groups, to be be precise) to get some hold on large scale patterns in these groups. After talking about how this connection works, I'll review some of the basic examples, going back even into the 1960s, then talk about the work of Hopkins-Miller-Behrens in the 2000s that uncovering some remarkable patterns using modular forms. This is only a start, and I'll end with some current vistas.