LEAPS-MPS: Fragments of Compactness

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。 最近的工作在nonelementary类表明，各种片段的紧性仍然可以保持在一些nonelementary类，仍然是强大的，足以证明各种结果的基本分类理论。 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。 该计划的第一年将带来一系列逻辑扬声器，专注于逻辑和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致性。 如上所述，紧性在非初等类中不成立，但PI将探索各种方法来找到这些类中的紧性片段。 第一个目标是将紧性和模型论的其他概念推广到高等代数。 第二个目标是发展非标准超积的非初等类使用范畴理论的观点。 第三个目标是继续PI的探索之间的连接非初等紧致性原则和大基数集theory.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.