LEAPS-MPS: Fragments of Compactness

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。最近关于非初等类的研究表明，在一些非初等类中，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。该计划的第一年将引入一系列逻辑学演讲者，专注于逻辑学与其他领域之间的联系。第二年将支持教师和学生在这些演讲者的启发下进行逻辑研究。更具体地说，PI将使用模型论、集合论和范畴论的技术研究非初等类的紧性。如上所述，紧凑性在非基本类中不成立，但是PI将探索各种方法来找到这些类中的紧凑性片段。第一个目标是将紧性和其他模型理论的概念扩展到高等代数。第二个目标是利用范畴论的观点发展非初等类的非标准超积。第三个目标是继续PI对集合论中非初等紧性原理与大基数之间联系的探索。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 nonelmentary classes has shown that various fragments of compactness can still hold in some nonelmentary 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). The first year of the program will bring in a series of logic speakers that focus on connections between logic and other fields. The second year will support both faculty and student research in logic inspired by these speakers.More specifically, the PI will research compactness properties in nonelementary classes using techniques from model theory, set theory, and category theory. As stated above, compactness fails to hold in nonelementary classes, but the PI will explore a variety of methods to find to fragments of compactness in these classes. The first objective is to extend compactness and other concepts of model theory to higher algebra. The second objective is to develop nonstandard ultraproducts in nonelementary classes using a category theoretic perspective. The third objective is to continue the PI’s exploration of connections between nonelementary compactness principles and large cardinal in set 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.