NSF-BSF: Independent Theories in Model Theory

模型论是数学逻辑的一个分支，它问我们对任意数学理论能说些什么。令人惊讶的是，在这样一个普遍的背景下，我们可以说出有意义的事情，一方面发现新的普遍定律，另一方面给出具体数学理论的新方法。模型理论分类理论提供了一种发现一般规律的方法：非正式地，它旨在隔离所谓的分界线，在那里可以检测到理论类别中的重大结构变化，并同时发展内部理论，以了解是什么产生了所观察到的现象。其中一条分界线与理论是否具有非平凡随机性有关。目前的方法在非随机方面在许多方面都是成功的。这个项目涉及独立的理论，即随机的理论。这种随机性在不同程度上使现有方法测量整个结构的能力复杂化。本研究使用一系列经典的和新开发的工具来检测模型理论随机性复杂性中的平均行为和变化，致力于对这些理论进行更基本的理解。随着研究的发展，该项目将支持各级学生的培训。更准确地说，该项目旨在研究具有独立性的独立理论。以下相关的测试问题指导了本项目的工作。第一个问题是描述简单理论的饱和模型，以一种类似于已知稳定理论的描述的方式，作为理解简单不稳定理论的组成部分的测试问题。解决这个问题的一种独立的方法涉及开发最近引入的分叉的泛化。第二个问题涉及到描述和发展随机图附近的一类有趣的简单理论的结构理论。第三种是研究Keisler阶商的结构。第四是在独立理论的范畴内找到真正的分界线，同时发展内部理论来证明这些是分界线。这些问题本身与理解所谓的凯斯勒秩序的经典问题有关，凯斯勒秩序在依赖理论上的结构是已知的，但在独立理论上的结构在某种意义上仍然是神秘的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Model theory, a branch of mathematical logic, asks what we can say about arbitrary mathematical theories. Surprisingly, for such a general context, we can say meaningful things, discovering new general laws on the one hand and giving new methods for concrete mathematical theories on the other. Model-theoretic classification theory gives an approach to discovering the general laws: informally, it aims to isolate so-called dividing lines where a significant structural change in the class of theories may be detected, and in parallel to develop the internal theory to understand what is producing the phenomena observed. One such dividing line has to do with whether theories have nontrivial randomness. Current methods have been successful in many ways on the non-random side. This project concerns independent theories, those on the random side. The randomness complicates to varying degrees the ability of current methods to gauge the full structure. This research works towards a more fundamental understanding of such theories, using a range of classical and newly developed tools able to detect average behavior and changes in the complexity of model-theoretic randomness. Along with the research developments, the project will support the training of students at various levels.More precisely, the project aims to study independent theories, those with the independence property. The following interrelated test problems guide the work in this project. The first concerns characterizing saturated models of simple theories in a way analogous to the known characterization for stable theories, as a test problem for understanding of the building blocks of simple unstable theories. An approach to this question, of independent interest, involves developing a recently introduced generalization of forking. A second problem involves characterizing and developing a structure theory for a certain interesting class of simple theories 'near' the random graph. A third is to investigate the structure of quotients of Keisler's order. A fourth is to find true dividing lines within the class of independent theories and in parallel to develop the internal theory justifying these as dividing lines. These questions themselves connect to the classical problem of understanding the so called Keisler order, whose structure on dependent theories is known but whose structure on the independent theories remains in some senses mysterious.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.