CAREER: Compactness in Incompact Worlds

模型理论领域研究结构的类别：群、场、图。这是一个广阔的研究领域，因此重要的区别是基于结构类如何描述(或公理化)。如果类可用一阶逻辑描述，则称其为初等类。一阶逻辑具有许多强大的性质，尤其是紧性。紧凑性使模型理论家能够构建具有奇异性质的结构，并推动了许多基本类的模型理论，最引人注目的是分类理论。然而，许多结构类不能用一阶逻辑描述(这些被称为非基本类)。由于缺乏紧凑性，非初等模型理论和分类理论的发展远远慢于初等模型理论和分类理论。最近在非初等类中的工作表明，紧性的各种片断在一些非初等类中仍然成立，并且仍然强大到足以证明初等分类理论的各种结果。PI将在非初级课程中开发更多这样的片段。此外，PI将运行一个计划，在他们的本土机构(R2机构和HSI)建立研究基础设施。这项计划将支持本科生进行逻辑研究，并由一系列演讲者支持，该系列演讲将在专家逻辑学家和教职员工和学生之间建立联系。这项研究将以各种方式发展紧凑性的片断。在范畴论中，我们将使用可及范畴和拓扑学中的方法来寻找某些非初等类的紧致性原理。借助于模型论和集合论，Erdos-Rado定理的广义不可分辨和推广将发现在所有非初等类中都成立的紧性原则。进一步利用集合论，大基数和紧致性原理(以及其他模型论思想)之间的联系将得到扩展。分类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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.