New model-theoretic structure

This project has two parts, both with the goal of investigating novel model-theoretic structure on mathematical objects. The first part concerns the étale-open topology and model theory of fields. The second part is to develop new model-theoretic notions of reducibility between first order structures and theories. First part: The goals are to obtain a better understanding of the étale-open topology both in general and in special cases, classify large fields according to properties of the étale-open topology, prove the large cases of conjectures from the model theory of fields, further develop tame topology over large fields, and investigate the connection between large fields and henselian rings. We work with general classes of fields, in particular large and éz fields, which contain all logically tame fields known before 2022. We will also develop a new and unexpected connection between large fields and henselian rings. Our approach combines algebraic geometry, commutative algebra, field theory, and model theory. This requires collaborators with an array of backgrounds. Second part: Previous classification-theoretic work concerns either individual theories or unary relations on the class of theories, e.g. stability. We consider new binary relations between theories, including trace definability and local trace definability. Trace definability is entirely novel but closely connected to known topics such as indiscernible collapse and Shelah expansions. We claim that it is a missing piece of the model- theoretic puzzle. The goals are to publish our work on trace definability, promote it within the model-theory community, and establish collaborations towards answering questions concerning it. We aim to answer basic questions concerning classification of theories modulo trace equivalence and trace definability of various algebraic structures. We often wish to show that one theory does not trace define another. Such problems can be approached via "brute force", but the best approach is to find new properties that are preserved under trace definability and separate theories.