Model theory is a subject which lies in the intersection of algebra and logic. Given a structure M, one can look for a set of axioms which completely pin down the (first-order) properties of M. For example, the field (C,+,⋅) is completely axiomatized by the axioms of algebraically closed fields of characteristic 0. To axiomatize a structure M, it is usually necessary to understand the “definable sets” of M, that is, the subsets D ⊆ Mn which are defined by (first-order) formulas. For example, in the field C, the definable sets are the constructible sets of algebraic geometry. The real exponential field (R,+,⋅,exp) has much more complicated definable sets. Nevertheless, it has a special property called “o-minimality” which implies that the definable sets have finite triangulations, and definable functions are piecewise smooth, among many other things.
In the process of axiomatizing structures and analyzing definable sets, model theorists have uncovered a number of “dividing lines” such as stability, NIP, simplicity, etc. These dividing lines reflect natural dichotomies in model theory. For example, if a theory T is unstable, then T has the maximum possible number of models, and if T is stable, then the models of T have a natural notion of “independence” similar to algebraic independence in C. The class of NIP theories generalizes the stable and o-minimal theories, includes many natural mathematical structures like R and Qp, and is closely connected to the concept of VC-dimension in machine learning.
My own research touches on each of these areas. I will discuss my partial results towards the classification of NIP fields and rings, as well as my work with Tran, Walsberg, and Ye on the model theory of large fields.
