Model Theory, Exponentiation and Quasiminimality

Jonathan Kirby

Model Theory is a way of looking at mathematical concepts which is sensitive to the choice of language we use to describe them. One of its achievements is to classify different mathematical theories according to their complexity. In this series of talks I will survey some of the work done towards understanding the model theory of exponential fields, including the real exponential and Zilber’s approach to the complex exponential field. We know from Wilkie that the real exponential field is not too complicated (it is o-minimal) and this has good consequences in geometry, in number theory, and even in machine learning. For the complex exponential, we do not know if it is tame (quasiminimal) or whether it is maximally complicated (interpreting both reals and integers). I will explain progress towards proving that it is tame.