Higher classification theory in model theory and applications

模型理论与应用中的高级分类理论

• 批准号: 2246598
• 负责人: Artem Chernikov
• 金额: $ 36万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246598&HistoricalAwards=false
• 关键词: Higher; classification; theory; model; applications

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.

模型理论研究了在某些受限制的形式语言中可以定义数学对象的方式，以及这些定义假设所暗示的结构属性。它提供了将有限结构的不对称问题转换为有关某些限制无限物体的形状，体积或维度的定性问题。这种研究方法起源于数学基础的问题，但近年来，它在研究古典数学和计算机科学的一些中心对象中发现了重要的应用。该项目进一步研究了这些联系，其主要动机是将现有技术从二进制结构（图）扩展到较高的ARITH（HyperGraphs）的结构，这代表了描述更复杂网络的数学方法，在该数学方式中，不仅在两个节点之间发生相互作用，而是在多个节点之间进行。这项研究将加深和扩展无限模型理论机制的应用范围，以在几何或代数上产生的超图中的组合中的问题，相反，对于组合剂在模型理论中的应用中的应用。该项目将涉及培训毕业生和本科生。Shelah的分类计划隔离了组合划分线（稳定性，歧视，nip等），分隔了数学结构，表现出各种程度的Gödelian行为，与驯服的理论相比，具有驯服的理论，即具有符合代数的代数的“几何学”理论。 Shelah的分类理论中的这些驯服说明通常是由对可定义二进制关系的组合复杂性的限制给出的。然后，如果限制在此分类的驯服侧，尤其是由各种代数或几何配置引起的图形，则可以显着改善图形组合中的许多中心结果。 PI将研究Shelah的分类理论的更高概括，在这种理论中，限制仅在较高的ARITH关系上，重点是N依赖性（n = 1对应于研究良好的NIP结构类别），N稳定性，N-稳定性，N-属性和N-Amalgamation，而不是传统的Binary Binary Case n = 1。这将应用于各种驯化结构（通过Keisler测量）中定义的超图的问题中的问题，以及对多项式扩展现象的概括（Elekes-Szabó型定理）的概括（elekes-szabó型定理），以及在诸如n-then-then-then-then-then-thersey the Is thersey theiss theiss theiss theS theS theS theS theS thiss thes thiss thisse theS的统计范围内的研究。通过基金会的智力优点和更广泛的影响评估标准通过评估来支持。