Topological dynamics in model theory

In our thesis we discuss the basis of a recent topic in logic, the use of topological dynamics in model theory. Mathematicians usually divide the universe of mathematical structures by their axioms: some mathematicians study groups, others study modules over rings, and so on. Model theorists divide this universe by classifying combinatorial properties of the definable sets. The class of the definable sets is the closure under finite unions, complements and projections of the very basic subsets of a structure (those defined by equations, inequalities, or any other primitive relation in the language). Of course, we can relate the two divisions of the universe to each other. However, the relation is not always trivial, because it might not be true that all structures satisfying a certain group of axioms share the same combinatorial properties of the definable sets. We will incidentally give examples of this phenomenon in the next section. Model theory has been very successful in the study of a class of structures which are called stable. Stable structures satisfy a (finite) combinatorial property that is reminiscent of the combination of Noetherianity and Artinianity in modules. For example, abelian groups and algebraic closed fields are stable structures, but other groups and fields are not. One of the main efforts of model theorists nowadays is to generalize methods from stable context to study unstable structures. The introduction of topological dynamics in model theory is a step in this direction. Topological dynamics studies G-flows, i.e. actions of a group G on a compact Hausdorff space, by homeomorphisms; the leading example is the flow of solutions of a differential equation. It is a well-established subject. It was introduced in model theory in 2009 by Ludomir Newelski, who aimed to generalize important notions of stable groups. Some examples of such generalizations will be provided in the following sections. A fundamental notion in model theory is that of a type, i.e. an intersection of definable subsets. We can provide a natural compact Hausdorff topology to the space of types, which can be seen as a flow. Generic types are intersections of large definable sets, called generic as well. A set is called generic if we can cover the whole group translating it finitely many times; it can be thought of as a generalization of a subgroup with a finite number of cosets. Unfortunately, generic types do not always exist in unstable context. Poizat amongst others showed that in stable context, generic types are connected with a group called G0. Newelski generalized these notions in an unstable context, introducing weakly generic types and showing their connection with another group, called G00. It is indeed a generalization because all these notions coincide in stable structures. These generalizations were provided using basic notions of topological dynamics, like minimal subflows of a G-flow.