RUI: Model Theory of Abstract Elementary Classes

这一提议是sarah 30年来为抽象初级类（AECs）开发分类理论计划的一部分。aec的定义和基本定理是由sheah在70年代提出的。AEC可以看作是一阶理论的一类模型的语义推广，它源于Jonsson在泛代数中的工作。2001年底，对《抽象小学类》分类理论的研究和关注大大增加。导致人们对非基本模型理论迅速产生兴趣的最具影响力的事件是Zilber试图理解Schanuel关于复数的猜想。Zilber引入了一类自然的模型，其中包含一个满足Schanuel猜想的指数函数的复数。虽然这个类不能通过一阶逻辑来管理，但它是一个AEC，它在所有不可数基数中都是绝对的，并且与Shelah的优秀类具有相同的属性。在平行发展中，Rami Grossberg和PI引入了驯服的概念，并在此假设下研究了伽罗瓦稳定的aec。在希拉的工作中，一个比温顺更弱的概念隐含地出现在分类aec的内部属性中。Grossberg和VanDieren进一步证明了aec分类理论中主要测试问题的类的一个特例，即Shelah的范畴猜想。本研究的目的是扩展非初等模型理论的程序，以改善一阶稳定性理论的知识体系，并发展一个自然框架，在这个框架内，逻辑以外的问题，例如代数几何，可以被解释和更好地理解。所提出的研究是在数学逻辑的一个分支模型论中进行的。模型理论家通常从一组公理（一个理论）开始，研究这些公理的不同解释或模型。最终目标是对理论进行分类，以预测模型的底层结构。这涉及到引入新的机制，如抽象的独立关系，然后可以用来回答其他数学分支的问题。模型论的大部分工作都局限于检验一阶逻辑所能表达的公理集。由于数学中的许多自然理论不允许在一阶逻辑中进行适当的处理，一阶逻辑的分类理论的应用领域受到限制。最近对代数几何和数论中的非一阶（非初等）例子的兴趣引发了对Shelah的非初等类分类理论计划的更多参与。提出的研究包括(1)证明非初等类别分类理论中主要测试问题的一个重要案例（Shelah’s Categoricity Conjecture）；(2)为一般非初等类别的tame aec发展一个稳定性理论；(3)促进代表性不足的群体参与数学。

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.