James Borger: Witt vectors and canonical lifts in families.

The Witt vector functor is an arithmetic construction that occurs in many arithmetic areas of algebra. But the basic definitions are often hard for newcomers to penetrate and even viewed by many experts as somewhat mysterious. In this talk, I'll explain a simple, conceptual construction of the functor and give a universal property, both of which deserve to be more widely known. (This approach was probably first found by Andre Joyal.) As an illustration of the usefulness of this perspective, I'll show how the canonical lift construction in the theory of elliptic curves in characteristic p>0 extends without effort to elliptic curves in families, and even families parameterized by p-adic formal schemes. I'll also explain how this allows for a purely formal proof of recent theorems of Erdogan and Finotti. (This observation is joint work with Lance Gurney.)