RUI: Model Theory of Abstract Elementary Classes

RUI：抽象基本类的模型理论

• 批准号: 0801313
• 负责人: Monica VanDieren
• 金额: $ 11.12万
• 依托单位: Robert Morris University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-05-01 至 2013-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801313&HistoricalAwards=false
• 关键词: RUI; Model; Theory; Abstract; Elementary

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.

该提案是 Saharon Shelah 长达 30 年的开发抽象基本类 (AEC) 分类理论计划的一部分。 AEC的定义和基本定理是Shelah在七十年代提出的。 AEC 可以被认为是一阶理论模型类的语义概括，根源于 Jonsson 在通用代数方面的工作。 到 2001 年底，抽象初级课程分类理论的活动和关注大幅增长。导致人们对非初等模型理论的兴趣快速增长的最有影响力的事件是齐伯试图理解尚纽尔对复数的猜想。 Zilber 引入了一类自然模型，其中包含具有满足 Schanuel 猜想的类似指数函数的复数。虽然这个类无法通过一阶逻辑来管理，但该类是一个 AEC，它在所有不可数基数中都是绝对的，并且与 Shelah 的优秀类共享属性。在并行开发中，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 AEC。 Shelah 的著作中隐含着一个比驯服更弱的概念，它是绝对 AEC 的内部属性。 Grossberg 和 VanDieren 继续证明了 AEC 分类理论中主要测试问题的驯服类别的特殊情况，即 Shelah 的类别性猜想。 这项研究的目的是扩展非初等模型理论的程序，改善一阶稳定性理论的知识体系，并开发一个自然框架，在该框架内可以解释和更好地理解逻辑之外的问题，例如代数几何中的问题。所提出的研究属于模型论，数理逻辑的一个分支。模型理论家通常从一组公理（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。这涉及引入新的机制，例如抽象独立关系，然后可以将其用于回答其他数学分支中的问题。模型论中的大多数工作仅限于检查可以使用一阶逻辑表达的公理集。 由于数学中有许多自然理论不允许在一阶逻辑中进行适当的处​​理，因此一阶逻辑分类论的应用领域是有限的。 最近对代数几何和数论中的非一阶（非基本）例子的兴趣引发了人们对 Shelah 的非基本类别分类理论计划的更多参与。拟议的研究包括（1）证明非初级类别分类理论中主要测试问题的一个重要案例（Shelah 的分类猜想），以及（2）为驯服的 AEC 的一般非初级背景开发稳定性理论，同时（3）促进代表性不足的群体参与数学。