Higher classification theory in model theory and applications

模型论研究数学对象可以用某种受限的形式语言来定义的方式，以及这些可定义性假设所隐含的结构属性。它提供了将有限结构的渐近问题转化为关于某些有限的无限物体的形状、体积或尺寸的定性问题的方法。这种研究方法起源于数学基础的问题，但近年来它在经典数学和计算机科学的一些中心对象的研究中发现了重要的应用。该项目进一步研究了这些连接，主要动机是将现有技术从二元结构（图）扩展到更高层次的结构（超图），超图代表了描述更复杂网络的数学方式，其中交互不仅发生在两个节点之间，而且同时发生在多个节点之间。这项研究将深化和扩大无限模型理论机器的应用范围，以几何或代数产生的超图的组合学问题，并反过来为组合学的应用，以开放的问题，在模型论。该项目将涉及研究生和本科生的培训。希拉的分类程序隔离组合分界线（稳定性，远端，NIP等）。分离数学结构表现出不同程度的哥德尔行为，从驯服的人，其中一个开发了一个“几何”理论类似于代数几何的可定义集在这样的结构。在希拉的分类理论中，这些驯服的概念通常是由对可定义的二元关系的组合复杂性的限制给出的。许多中央结果图组合然后可以大大改善，如果一个限制图的驯服方面，这一分类，特别是图所产生的各种代数或几何配置。PI将研究Shelah分类理论的更高概括，其中限制仅放在更高的arity关系上，专注于n-依赖（n=1的情况对应于研究良好的NIP结构类），n-稳定性，n-远端性和n-融合，而不是传统的二元情况n=1。这将适用于在极端组合的超图定义在各种驯服结构的问题（通过Keisler措施），以及多项式展开现象的推广（Elekes-Szabó型定理），和代数结构的研究，如群和领域的定义，在n-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进行评估，被认为值得支持审查标准。

Model theory studies the ways in which mathematical objects can be defined in some restricted formal language, and what structural properties are implied by these definability assumptions. It provides methods of converting asymptotic questions about finite structures into qualitative questions about the shape, volume or dimension of certain limiting infinite objects. This method of study originated in questions on foundations of mathematics, but in recent years it has found important applications in the study of some central objects of classical mathematics and computer science. The project investigates further these connections, with the major motivation of extending the existing techniques from binary structures (graphs) to structures of higher arity (hypergraphs), which represent a mathematical way of describing more complex networks in which interactions happen not just between two nodes at a time, but between multiple nodes simultaneously. This study will both deepen and extend the scope for applications of the infinitary model-theoretic machinery to questions in combinatorics of geometrically or algebraically arising hypergraphs, and conversely for applications of combinatorics to open questions in model theory. The project will involve training of graduate and undergraduate students.Shelah's classification program isolates combinatorial dividing lines (stability, distality, NIP, etc.) separating mathematical structures exhibiting various degrees of Gödelian behavior, from the tame ones in which one develops a “geometric” theory akin to algebraic geometry for definable sets in such structures. These tameness notions in Shelah’s classification theory are typically given by restrictions on the combinatorial complexity of definable binary relations. Many of the central results in graph combinatorics can be then improved dramatically if one restricts to graphs on the tame side of this classification, in particular to graphs arising from various algebraic or geometric configurations. The PI will investigate a higher generalization of Shelah's classification theory, where the restriction is only put on higher arity relations, focusing on n-dependence (with the case n=1 corresponding to the well studied class of NIP structures), n-stability, n-distality, and n-amalgamation, as opposed to the traditional binary case n=1. This will be applied to questions in extremal combinatorics of hypergraphs definable in various tame structures (via Keisler measures), as well as to generalizations of the polynomial expansion phenomena (Elekes-Szabó type theorems), and to the study of algebraic structures such as groups and fields definable in n-tame theories.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.