CAREER: Compactness in Incompact Worlds

模型论领域研究结构类：群、域、图。这是一个广泛的研究领域，因此重要的区别是基于如何描述（或公理化）结构类。如果一个类可以在一阶逻辑中描述，则称之为初等类。一阶逻辑具有许多强有力的性质，特别是紧性。紧性允许模型理论家建立具有奇异性质的结构，并推动了许多基本类的模型理论，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）。由于缺乏紧致性，非初等模型论和分类论的发展比初等模型论和分类论的发展要慢得多。最近在非初等类中的工作表明，紧性的各种片段仍然可以在某些非初等类中成立，并且仍然足够强大，可以证明初等分类理论的各种结果。PI将在非初等类中开发更多的这些片段。此外，PI将在其所在机构（R2机构和HSI）开展一项建立研究基础设施的计划。 该计划将支持本科生进行逻辑研究，由一系列演讲者支持，这些演讲者将建立专家逻辑学家与教师和学生之间的联系。这项研究将以各种方式开发紧凑性的片段。 从范畴论，方法从可及范畴和从拓扑将被用来找到紧凑性原则，在某些非初等类。 利用模型论和集合论，广义不动点定理和埃尔多-拉多定理的推广将找到在所有非初等类中成立的紧性原理。 进一步借鉴集合论，大基数和紧性原理（沿着与其他模型论思想）之间的联系将得到扩展。 该奖项反映了NSF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.