In his influential paper “Tameness in expansions of the real field” from the early 2000s, Chris Miller wrote:
“ What might it mean for a first-order expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and real-analytic geometers to the o-minimal setting: expansions of the real field in which every definable set has finitely many connected components. But there are expansions of the real field that define sets with infinitely many connected components, yet are tame in some well-defined sense […]. The analysis of such structures often requires a mixture of model-theoretic, analytic-geometric and descriptive set-theoretic techniques. An underlying idea is that first-order definability, in combination with the field structure, can be used as a tool for determining how complicated is a given set of real numbers.”
Much progress has been made since then, and in this talk I will discuss an updated account of this research program. I will consider this program in the larger generality of expansions of the real ordered additive group (rather than just in expansions of the real field as envisioned by Miller). In particular, I will mention in this context recent joint work with Erik Walsberg, in which we produce an interesting tetrachotomy for such expansions.