Higher classification theory in model theory and applications

模型理论研究数学对象可以用一些有限的形式语言来定义的方法，以及这些可定义性假设所隐含的结构特性。它提供了将有限结构的渐近问题转化为关于某些极限无限物体的形状、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问题，但近年来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，其主要动机是将现有技术从二元结构（图）扩展到更高密度的结构（超图），这代表了一种描述更复杂网络的数学方法，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。本研究将深化和扩展无限模型论机制的应用范围，以解决几何或代数上产生的超图的组合问题，反过来，将组合学应用于模型论中的开放问题。该项目将包括研究生和本科生的培训。Shelah的分类程序分离了组合分界线（稳定性，远性，NIP等），将表现出不同程度Gödelian行为的数学结构从温和的数学结构中分离出来，其中人们开发了类似于代数几何的“几何”理论，用于此类结构中的可定义集。希拉分类理论中的这些驯服概念通常是由可定义二元关系的组合复杂性的限制给出的。如果将图组合学中的许多中心结果限制在这种分类的温和方面，特别是由各种代数或几何构型产生的图，则可以显着改进。PI将研究Shelah分类理论的更高泛化，其中限制只放在更高的密度关系上，关注n-依赖性（情况n=1对应于已经研究好的NIP结构类），n-稳定性，n-远性和n-合并，而不是传统的二元情况n=1。这将应用于在各种驯服结构中可定义的超图的极值组合问题（通过Keisler测度），以及多项式展开现象的推广（Elekes-Szabó类型定理），以及代数结构的研究，例如在n-驯服理论中可定义的群和域。该奖项反映了美国国家科学基金会的法定使命，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。

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.