This proposal is part of Saharon Shelah's 30 year old program of developing a classification theory for Abstract Elementary Classes (AECs). The definition of AECs and the basic theorems were introduced by Shelah in the seventies. An AEC can be thought of as a semantic generalization of the class of models of a first order theory and has roots in work of Jonsson in universal algebra. Toward the end of 2001 activity in and attention to classification theory of Abstract Elementary Classes grew substantially. The most influential event leading to the fast-growing interest in non-elementary model theory was Zilber's attempt to understand Schanuel's conjecture over the complex numbers. Zilber introduced a natural class of models containing the complex numbers with an exponentiation-like function that satisfies Schanuel's conjecture. While this class was not manageable by first order logic, the class is an AEC which is categorical in all uncountable cardinals and shares properties with Shelah's excellent classes. In a parallel development, Rami Grossberg and the PI introduced the notion of tameness and studied Galois-stable AECs under this assumption. A weaker notion than tameness appeared implicitly in Shelah's work as an internal property of categorical AECs. Grossberg and VanDieren went on to prove a special case for tame classes of the main test question in the classification theory for AECs, namely Shelah's Categoricity Conjecture. The purpose of this investigation is to expand this program of non-elementary model theory with the effects of improving the body of knowledge of first order stability theory and developing a natural framework within which problems outside of logic, for instance in algebraic geometry, can be interpreted and better understood.
The research proposed is in model theory, a branch of mathematical logic. A model theorist typically starts with a set of axioms (a theory) and studies the different interpretations or models of these axioms. The ultimate goal is to classify the theories to predict the underlying structure of the models. This involves introducing new machinery such as abstract independence relations which can then be used to answer problems in other branches of mathematics. Most work in model theory has been confined to examining sets of axioms which can be expressed using first-order logic. Since there are many natural theories in mathematics that do not admit an appropriate treatment in first-order logic, the application field of the classification theory for first-order logic has limits. Recent interest in non-first-order (non-elementary) examples from algebraic geometry and number theory has triggered increased involvement in Shelah's program of classification theory for non-elementary classes. The proposed research involves (1) proving an important case of the main test-question in the classification theory for non-elementary classes (Shelah's Categoricity Conjecture) and (2) developing a stability theory for the general non-elementary context of tame AECs, while (3) facilitating under-represented group participation in mathematics.
