Model-theoretic tree properties and their applications

模型理论树的性质及其应用

• 批准号: 2246992
• 负责人: Samuel Ramsey
• 金额: $ 19.5万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard 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=2246992&HistoricalAwards=false
• 关键词: Model; theoretic; tree; properties; their

Model theory is a branch of mathematical logic dedicated to studying definable sets in mathematical structures. Model theorists aim to isolate certain combinatorial features enjoyed by definable sets in tame structures and to develop a general theory of structures satisfying these properties. This unifies and explains the tameness of a structure or family of structures, but also provides an engine for proving deep structure theorems by exploiting the combinatorial similarity of a structure under analysis to well-understood mathematical objects. The model-theoretic notion of simplicity was a distillation of the tameness of sets definable in random graphs, pseudo-finite fields, and algebraically closed difference fields and has served as a key ingredient in applications of model theory to combinatorics and algebraic dynamics. Simplicity was the first of the model-theoretic tree properties to be isolated and studied, but since then there have been a number of model-theoretic properties that have received intensive study, generalizing and deepening the achievements of simplicity theory. The PI will continue to develop the theory of model-theoretic tree properties and pursue applications in algebra and combinatorics. The project also provides research training opportunities for graduate students. The simple theories were defined by Shelah as the class of theories without the tree property, a combinatorial property of a formula that he isolated in the study of independence in stable theories. The first applications were set-theoretic, but the core tools of simplicity theory also led to a consolidation of the techniques at play in the analysis of concrete structures coming from algebra and combinatorics, most notably in the case of pseudo-finite fields and smoothly approximable structures built out of classical geometries over finite fields. The development of the theory, then, sparked a two-way exchange between the theory and the examples, leading to refined understanding of core notions in model theory. The template established by simple theories, of a theory for class of theories defined by a combinatorial condition on trees of definable sets built on a theory of independence, was successfully replicated for a variety of dividing lines. Moreover, this extension has revealed striking connections and applications to combinatorics, allowed for the analysis of core new algebraic examples, and led to the resolution of difficult open problems in model theory. This project takes a systematic approach to model-theoretic tree properties, consolidating the core techniques to address both internal questions within model theory and applications to a diverse array of structures coming from algebra, geometry, and combinatorics.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.

模型理论是数学逻辑的一个分支，致力于研究数学结构中的可定义集。 模型理论家旨在隔离驯服结构中可定义的集合所享有的某些组合特征，并开发满足这些特性的结构的一般理论。这统一并解释了结构或结构系列的驯服性，但也提供了一种引擎，通过利用正在分析的结构的组合相似性来证明深层结构定理的引擎。 简单性的模型理论概念是对在随机图，伪限制字段和代数闭合差异领域中可定义的集合的腐烂性的提炼，并已在将模型理论应用于组合剂和代数动力学中是一种关键成分。 简单性是要隔离和研究的模型理论树特性的第一个，但是从那时起，就已经有许多模型理论特性接受了深入的研究，概括和加深了简单理论的成就。 PI将继续发展模型理论树特性的理论，并在代数和组合中追求应用。该项目还为研究生提供了研究培训机会。简单的理论由谢拉（Shelah）定义为没有树特性的理论类别，这是他在稳定理论中独立研究中隔离的公式的组合特性。 最初的应用是设定的，但是简单理论的核心工具也导致了在分析来自代数和组合学的混凝土结构中的技术的巩固，最值得注意的是，在伪金融领域以及在有限的几何场上建立的经典遗传构建的伪造型结构。 因此，该理论的发展引发了理论与示例之间的双向交流，从而使对模型理论中核心概念的理解得到了精致的理解。由简单理论建立的模板，是一种基于独立理论的可定义集的树木的组合条件定义的一类理论的理论，已成功地用于各种分界线。 此外，这一扩展已揭示了与组合学的惊人连接和应用，可以分析核心新代数示例，并导致模型理论中的困难开放问题解决了。 该项目采用一种系统的方法来建模树木的特性，巩固了核心技术，以解决模型理论中的内部问题，并应用于来自代数，几何学和组合学的各种结构中的应用。这奖反映了NSF的立法任务，并被认为是通过基金会的智力效果和广泛的评估来评估的，并值得通过评估来进行评估。