Dividing lines among fields

Abstract

In this article we will discuss some results on the classification of stable, simple, and NIP fields from the perspective of model theory. Additionally, we will explore recent work by Silvain Rideau-Kikuchi and the author, where a general class of fields (pseudo $I$-closed fields) was constructed providing new examples of NTP$_2$ fields. These fields satisfy a \emph{local-global} principle for the existence of points on varieties with respect to models of certain theories. Finally, we will address open questions and outline future research directions