CAREER: Compactness in Incompact Worlds

模型论领域研究结构类：群、场、图。这是一个广泛的研究领域，所以重要的区别是基于如何描述（或公理化）这类结构。如果一个类在一阶逻辑中是可描述的，它就被称为基本类。一阶逻辑具有许多强大的性质，特别是紧性。紧凑性允许模型理论家构建具有奇异性质的结构，并推动了许多初级类的模型理论，最著名的是分类理论。然而，许多结构类在一阶逻辑中是不可描述的（这些被称为非初等类）。由于缺乏紧凑性，非初等模型理论和分类理论的发展比初等模型理论的发展要慢得多。最近对非初等类的研究表明，在某些非初等类中，各种紧性片段仍然存在，并且仍然足以证明初等分类理论的各种结果。PI将在非小学课程中开发更多这样的片段。此外，PI将运行一个项目，在他们的家乡机构（R2机构和HSI）建立研究基础设施。该计划将支持本科生进行逻辑研究，由一系列演讲者支持，将建立专家逻辑学家与教师和学生之间的联系。本研究将以多种方式发展紧凑性片段。从范畴论出发，利用可访问范畴和拓扑的方法来寻找某些非初等类的紧性原理。利用模型论和集合论，Erdos-Rado定理的广义不可分辨性和一般化将找到所有非初等类都适用的紧性原理。进一步利用集合论，将扩展大基数和紧性原则（以及其他模型理论思想）之间的联系。在范畴逻辑中形成的联系将扩展到更高的范畴理论，为模型理论开辟新的领域。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。

The field of model theory studies classes of structures: groups, fields, graphs. This is a broad field to study, so important distinctions are made based on how the class of structures is described (or axiomatized). If the class is describable in first-order logic, it is called an elementary class. First-order logic has many powerful properties, especially the property of compactness. Compactness allows model theorists to build structures with exotic properties and has driven much of the model theory of elementary classes, most notably classification theory. However, many classes of structures are not describable in first-order logic (these are called nonelementary classes). Lacking compactness, the development of nonelementary model theory and classification theory has proceeded much slower than its elementary counterpart. Recent work in nonelementary classes has shown that various fragments of compactness can still hold in some nonelementary classes and are still powerful enough to prove various results of elementary classification theory. The PI will develop more of these fragments in nonelementary classes. Additionally, the PI will run a program to build research infrastructure at their home institution (an R2 institution and HSI). This program will support undergraduates conducting research in logic, supported by a speaker series that will build connections between expert logicians and faculty and students.This research will develop fragments of compactness in a variety of ways. From category theory, methods from accessible categories and from topoi will be used to find compactness principles in certain nonelementary classes. Drawing on model theory and set theory, generalized indiscernibles and generalizations of the Erdos-Rado theorem will find compactness principles that hold in all nonelementary classes. Further drawing on set theory, the connections between large cardinals and compactness principles (along with other model-theoretic ideas) will be extended. The connections formed in categorical logic will be extended to higher category theory to open new areas for model theory.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.