| Résume | This will be a personal talk surveying some of my work with Zoé Chatzidakis over the last three decades. Mathematically it forms a rather coherent chapter in the basic model theory of difference equations, combining ideas from geometric stability and simplicity, model-theoretic algebra and algebraic geometry.
The signposts include axiomatization, two theories of dimension, higher amalgamation, elimination of imaginaries, stable embeddedness; a structure theorem for the basic geometries, taking the form of a trichotomy; beyond finite dimensional difference varieties, a stationarity theorem; and applications to algebraic dynamics. To the extent that time permits, I will discuss a forthcoming joint result refining the trichotomy a little: finite-dimensional difference varieties admit dévissage to a combination of one-dimensional ones, and ones coding the dynamics of multiplication by a fixed group element in an algebraic homogeneous space, and a soon to be published work of Zoé's on the structure of locally modular groups.
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