I will discuss the Hanna Neumann conjecture of the 1950's and some tools in graph theory that I used to solve it.  The tools include sheaf theory on graphs, Galois theory for graphs, and the preservation of "local properties" under base change (for graphs).  Time permitting, I will mention one or two other applications of sheaves on graphs.This talk assumes only basic graph theory and linear algebra.  This talk will not mention any number theory or algebraic geometry per se, except possibly Chevalley's Theorem on constructible sets.