Mathematics Department Colloquium, April 16, 1998

Ehud Hrushovski, Dept. of Mathematics, Hebrew University, Jerusalem; and MSRI.

Is there a geometry of difference equations?

A difference equation is an algebraic equation in several variables, involving an additional symbol σ intended to denote a field automorphism. A difference variety is the set of solutions (in a field with a distinguished automorphism) to a system of difference equations.

A classical example is the equation σ(x) = x, coupled with algebraic equations f(x) = 0 defining a variety V. If σ is taken to be the automorphism x ↦ x^{p} in a field of characteristic p, the corresponding difference variety coincides with the set of points of V over the finite field GF(p). If σ is the operator f(x) ↦ f(x + 1) on a field of functions, the solutions are the periodic maps into V.

I will try to describe the general geography of the category of difference varieties. It contains (in two ways) the category of algebraic varieties. A number of model theoretic results suggest that the most interesting sites lie in a small but proper neighborhood of those described by algebraic geometry.