Étale methods in model theory

In 1989, Van den Dries showed that definable sets, in henselian valued fields of characteristic zero, admit a particularly nice description, that somehow resembles the one available for definable sets in algebraically closed fields. A result like this one begs the question -- can something similar be said about other classes of fields? Can one find a unifying framework encompassing all fields (possibly with some further hypotheses) whose definable sets can be decomposed this way? The first step is, necessarily, finding an adequate notion of "open" -- such a notion should, then, survive a reality check: it should coincide with available, classical notions of openness, like Zariski open or valuation open, in all natural cases. In come étale methods. Or more specifically, the étale-open topology, which is a candidate for the notion of "openness" that was needed. One might think of it as a way of assigning, uniformly, a topology to all k-varieties over a certain field k; and indeed, it is the data of a covariant functor that carries over geometrical (and algebraic) information from the k-varieties to certain topological spaces obtained from their k-points. The étale-open topology is two-faced -- on the one hand, it$ acts as a dictionary between algebraic properties of k and topological properties. As an example, the field k is not separably closed if and only if this topology is Hausdorff on every quasi-projective k-variety. On the other, it generalizes several well-known topologies: if k is separably closed, for example, it is just the Zariski topology; if k is real closed, it is the order topology; if it is henselian, it is the valuation topology. Bothof these faces come into play when the étale-open topology is applied to obtain results on the algebraic properties of certain model-theoretically interesting fields. In chapter 3 and 4, two of these kind of applications are explored -- under the assumption of largeness, without which the étale-open topology is just the discrete topology: a specific instance of the Stable Fields Conjecture is proved, namely that stable large fields are separably closed; and on a similar vein, in the following chapter, simple large fields are shown to be bounded. Finally, chapter 5 goes back to the beginning of our story. Now that we have a notion of "openness" at hand, we can isolate the class of fields where definable sets admit a nice topological decomposition, namely éz fields. In particular, two examples of éz fields are explored in detail: henselian valued fields of characteristic zero, and algebraically maximal Kaplansky valued fields. All of the work is based on papers by Johnson, Tran, Walsberg and Ye.