Dr Isabel Müller (Imperial College London) gives a maths seminar at the School of Mathematics, UEA. The talk is on “The Free Group and further Non-equational Stable Groups”. The abstract is below.

Like many concepts in model theory, also the notion of equationality bases its intuition in algebra. More specifically, it is an abstraction of noetherianity into the logic context: In algebraically closed fields, instances of first order formulas are boolean combinations of varieties, i.e. Zariski closed sets. These are noetherian, meaning that an infinite intersection of varieties is already given by a finite subintersection. Transferring this principle to model theory, we say that a first order formula is equational if any infinite intersection of its instances is equivalent to a finite subintersection. We then call a theory equational, if any formula is the boolean combination of equational formulas.

In the classification of first order theories, the class of stable theories plays a fundamental role. An easy proof shows that any equational theory is necessarily stable. The converse question is more complex. Until recently, the only known natural example of a stable, non-equational theory was given by the non-abelian free group. The proof by Zlil Sela relies on deep geometric tools and was not accessible to the community of model theorists. We will present a new criterion for the non-equationality of a theory, which yields a short elementary proof of the non-equationality of the free group and generalizes to the larger class of free products of stable groups. That indicates that the difference between equational and stable theories is much larger then previously assumed.

We will introduce all notions stemming from Model Theory and try not to assume knowledge in first order logic. This is joint work with Rizos Sklinos.