CAREER: Model Theory and Homogeneous Structures

模型论是数理逻辑的一个分支，研究数学结构(即具有一定运算的集合)的组合性质。由于大多数数学对象可以表示为一个结构，它的主题非常广泛，它与数学的几个领域相互作用。然而，其他数学领域对特定的结构或结构类别感兴趣(例如，算术研究带有加法和乘法的整数的结构)，而模型理论则退一步，研究一般的结构，寻找分界线和共同性质。它挑出了由组合驯服条件定义的大类结构(这意味着结构在某种意义上并不太复杂)，然后试图理解这些类中的结构。其中一类是NIP结构，它可以被认为是具有几何味道的结构。在过去的十年里，这个课程受到了很大的关注。另一类是同质结构，它是具有许多对称性的结构。这些都是属于组合学的对象，在模型理论之外或在模型理论的边缘进行了大量研究。这个项目的主要目标是研究新的模型论工具来理解同质结构，特别是那些NIP结构。关于均匀结构的一般定理很少，同时也没有证据表明不可能有任何定理：我们不知道均匀结构可以有多复杂，本项目希望阐明这一点。本项目的起点是PI获得的原始秩1 NIP均匀结构的分类。一个目标自然是将这一结果推广到所有有限秩NIP齐次结构。更长期的目标是对所有NIP同质结构进行分类。这将需要理解omega分类线性顺序，这本身就是一个有趣的项目。该项目的一个更普遍的目标是开发研究同质结构的模型理论工具。这将建立在上面提到的工作以及之前与Itay Kaplan关于有限生成的自同构群的稠密子群的工作的基础上。项目包括找到Ramsey性质的模型理论结果并澄清其与远程性的联系，找到具有良好性质的齐次展开式允许的充要条件(包括具有平稳或规范的独立关系、线性有序、Ramsey等)。最后，一个更具投机性的项目是对同质结构中类似随机行为的理解，并找到测量同质结构的复杂性的方法，特别是那些位于通常的模型理论TAME类之外的结构。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。

Model theory is a branch of mathematical logic that studies combinatorial properties of mathematical structures (that is, sets equipped with certain operations). Since most mathematical objects can be represented as a structure, its subject matter is very broad and it interacts with several areas of mathematics. However, whereas other fields of mathematics are interested in specific structures, or classes of structures (for instance arithmetic studies the structure of the integers with addition and multiplication), model theory takes a step back and studies structures in general, looking for dividing lines and common properties. It singles out large classes of structures defined by combinatorial tameness conditions (which imply that the structure is in some sense not too complicated) and then attempts to understand the structures in those classes. One such class is that of NIP structures, which can be thought of as structures that have a geometric flavor. This class has received a lot of attention in the past decade. Another class is that of homogeneous structures, which are structures that have a lot of symmetries. Those are objects that belong to combinatorics and have been largely studied outside, or at the border of model theory. The main goal of this project is the investigation of new model-theoretic tools to understand homogeneous structures and in particular those that are NIP. There are few general theorems on homogeneous structures and at the same time no evidence that there cannot be any: we do not know how complicated homogeneous structures can be and this project hopes to shed light on this.The starting point of this project is the classification of the primitive rank 1 NIP homogeneous structures obtained by the PI. One goal is naturally to generalize this result to all finite rank NIP homogeneous structures. A more long-term goal is the classification of all NIP homogeneous structures. This would require understanding omega-categorical linear orders, which is an interesting project in its own right. A more general goal of the project is the development of model-theoretic tools to study homogeneous structures. This will build on the work mentioned above as well as on a previous work with Itay Kaplan on finitely generated dense subgroups of automorphism groups. Projects include finding model-theoretic consequences of the Ramsey property and clarifying its link with distality, finding necessary and sufficient conditions for admitting homogeneous expansions with nice properties (including having a stationary or canonical independence relation, being linearly ordered, Ramsey etc.). Finally, a more speculative project is the understanding of random-like behaviors in homogeneous structures and finding ways to measure the complexity of homogeneous structures, in particular those that lie outside the usual model-theoretic tame classes.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.