NSF-BSF: Independent Theories in Model Theory

模型理论是数理逻辑的一个分支，它询问我们可以对任意的数学理论说些什么。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，一方面发现新的一般规律，另一方面为具体的数学理论提供新的方法。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，它旨在分离可能在理论类别中发现重大结构变化的所谓分界线，并同时发展内部理论，以理解是什么产生了所观察到的现象。一个这样的分界线与理论是否具有不平凡的随机性有关。目前的方法在许多方面都在非随机方面取得了成功。这个项目涉及独立的理论，那些随机的理论。这种随机性在不同程度上使当前方法测量完整结构的能力变得复杂。这项研究致力于更基本地理解这类理论，使用一系列经典和新开发的工具，能够检测平均行为和模型理论随机性复杂性的变化。随着研究的发展，该项目将支持各级学生的培养。更准确地说，该项目旨在学习独立的理论，那些具有独立性质的理论。以下相关的测试问题指导了本项目的工作。第一个涉及以类似于稳定理论的已知特征的方式来描述简单理论的饱和模型，作为理解简单不稳定理论的基础的测试问题。对这个问题的一种独立关注的方法，包括开发最近引入的分叉的泛化。第二个问题涉及为随机图附近的某一类有趣的简单理论刻画和发展结构理论。第三个是研究凯斯勒阶商的结构。第四是在独立的理论中找到真正的分界线，同时发展内部理论，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，该秩序关于依赖理论的结构是已知的，但其关于独立理论的结构在某种意义上仍然是神秘的。这一裁决反映了NSF的法定使命，并通过使用基金会的智力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.