Model-theoretic tree properties and their applications

模型论是数理逻辑的一个分支，致力于研究数学结构中的可定义集合。 模型理论家的目标是分离出驯服结构中的可定义集合所具有的某些组合特征，并发展出满足这些性质的结构的一般理论。这统一并解释了结构或结构族的驯服性，但也提供了一个引擎，通过利用分析下的结构与理解良好的数学对象的组合相似性来证明深层结构定理。 模型论的简单性概念是对随机图、伪有限域和代数闭差域中可定义的集合的驯服的升华，并且是模型论在组合学和代数动力学中应用的关键成分。 简单性是第一个被分离和研究的模型理论树属性，但从那时起，已经有一些模型理论属性得到了深入的研究，推广和深化了简单性理论的成就。 PI将继续发展理论的模型理论树的属性，并追求在代数和组合学的应用。该项目还为研究生提供研究培训机会。简单的理论被定义为一类理论的希拉没有树的财产，一个组合财产的一个公式，他孤立的研究独立稳定的理论。 最早的应用是集合论，但简单性理论的核心工具也导致了在分析来自代数和组合学的具体结构时发挥作用的技术的巩固，最明显的是在伪有限域和有限域上经典几何的光滑近似结构的情况下。 然后，理论的发展引发了理论和实例之间的双向交流，导致对模型理论中核心概念的精细理解。模板建立了简单的理论，理论类的理论定义的组合条件树的可定义的集合建立在理论的独立性，成功地复制了各种分界线。 此外，这种扩展揭示了惊人的连接和应用组合，允许分析核心新的代数例子，并导致解决困难的开放问题的模型论。 该项目采用系统的方法来研究模型理论树的属性，巩固核心技术，以解决模型理论内部的问题，并将其应用于代数、几何和组合学等多种结构。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.