Nadav Meir (Imperial College London) gives a maths seminar at the School of Mathematics, UEA. The talk is on “O-minimality, pseudo-o-minimality: on first-order properties of semialgebraic sets”. The abstract is below.

A semialgebraic set is a set given by the real solutions to a finite set of polynomial equations and inequalities. In Grothendieck’s Esquisse d’un Programme he suggested the following challenge: Investigate classes of sets with the tame topological properties of semialgebraic sets. O-minimality is model theory’s response to Grothendieck’s challenge; it is a property of ordered structures exhibiting the “tame” topological properties of semialgebraic sets, such as cell decompositions and stratifications. With applications both within and outside of model theory, the notion of an o-minimal structure has proven to be increasingly useful, with applications varying from real algebraic and real analytic geometry to economics and machine learning.

An elementary result on o-minimal structures states that any structure satisfying the same first-order sentences as an o-minimal structure is itself o-minimal. Despite that fact, there is no axiomatization of o-minimality by a set of first-order sentences; this can be seen by taking ultraproducts, as we will see.

In this talk, we will review the definition and key results of o-minimality. We will then survey a few first-order properties of semialgebraic sets, each property “tame” in its own way, generalizing the tameness of o-minimal structures. We will end by discussing structures satisfying all first-order properties which hold in every o-minimal structure, what tameness properties these structures satisfy and how they can be axiomatized.