Let P be a poset and let Q be a partially ordered topological space. Then P^Q denotes the set of (continuous) order-preserving maps from Q to P, ordered pointwise, where P has the discrete topology. (In 1942, Birkhoff conjectured that if P, Q, and R are finite, non-empty posets such that P^Q is isomorphic to R^Q, then P is isomorphic to R.)
If P^Q is a lattice, then we call it a "function lattice." Expanding on a survey of Davey and Duffus, we will show how duality theory gives one a tool, called a logarithm, for understanding the structure of function lattices. If time permits, we will also discuss McKenzie's recent progress towards solving Birkhoff's conjecture.