Normal functions are to families of algebraic cycles what period maps are to families of algebraic varieties. They are given by integrals of differential forms over non-closed chains instead of topological cycles, and satisfy inhomogeneous differential equations instead of homogeneous ones. The simplest examples are sections of families of elliptic curves, and I will spend some time discussing such an example. After that I will give a general picture of how they arise, what the main current problems about them are, and state a few recent results.
Vladimir Rubtsov, Vasily Golyshev, Ilia Gaiur
