Model Theoretic Classification Theory and Finite Combinatorics

模型理论分类理论和有限组合学

基本信息

批准号：
2115518
负责人：
Caroline Terry
金额：
$ 16.43万
依托单位：
Ohio State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-01-15 至 2023-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2115518&HistoricalAwards=false
关键词：
Model Theoretic Classification Theory Finite

项目摘要

Model theory is a branch of mathematical logic which seeks to understand common structural phenomena driving the behavior of different types of mathematical objects. A crucial idea in this area, first developed in the 1970s by Shelah, is the notion of a dividing line. A dividing line can be thought of as a structural dichotomy within a certain class of mathematical objects. Many of the most important dividing lines correspond to local combinatorial properties which have significant implications for global structure. In the infinite setting, model theorists have had great success using dividing lines to classify examples and generalize their behavior. However, extensions into the finite setting have been limited, largely due to the failure there of crucial infinitary tools. On the other hand, extremal and arithmetic combinatorics are fields which focus on the finite setting, but which study many of the same themes as model theory, such as local versus global structure and the interplay of structure and randomness. These fields have developed finitary questions and tools which are new to model theory, but which have have deep connections to model theoretic ideas. The goal of this project is to extend the study of model theoretic dividing lines in the finite setting by solving finitary problems from extremal and additive combinatorics which address these shared themes.More specifically, this project will focus on finding local model theoretic conditions which have robust implications for bounds and growth rates in theorems from additive and extremal combinatorics. This will be accomplished in two main directions. The first will focus on questions from additive combinatorics. Here a main goal will be to identify structural dichotomies for subsets of high-dimensional vector spaces over prime fields. For instance, what kinds of sets are most "tame'', as measured through improved bounds in structural decomposition theorems? Can the "tame'' sets be characterized by local combinatorial configurations? The second direction will address questions in extremal combinatorics. Specifically, the PI will continue work on enumeration and extremal problems for hereditary properties in finite relational languages.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，它试图理解推动不同类型数学对象的行为的常见结构现象。在1970年代，谢拉（Shelah）最初开发的这一领域的一个关键想法是分裂线的概念。可以将分界线视为某些数学对象中的结构性二分法。许多最重要的分界线对应于局部组合特性，这些特性对全球结构具有重要意义。在无限的环境中，模型理论家使用分隔线来对示例进行分类并概括其行为取得了巨大的成功。但是，有限设置的扩展是有限的，这主要是由于关键的无限工具失败。另一方面，极端和算术组合是集中在有限设置上的领域，但是研究了许多与模型理论相同的主题，例如局部与全局结构以及结构和随机性的相互作用。这些领域已经开发了对模型理论的新问题和工具，但它们与建模理论思想具有深厚的联系。该项目的目的是通过解决解决这些共享主题的极端和添加剂组合学的有限问题，扩展对有限设置中模型理论分裂线的研究。具体来说，该项目将集中于寻找本地模型理论条件，这些理论条件对添加剂和极端组合剂对Theorems的界限和增长率具有强大的含义。这将在两个主要方向上完成。第一个将重点关注添加剂组合学的问题。在这里，主要目标是确定主要田地上高维矢量空间子集的结构二分法。例如，通过结构分解定理中的改进范围来衡量哪些集合最“驯服”？“驯服”集可以以局部组合配置为特征？第二个方向将解决极端组合学中的问题。具体而言，PI将继续在有限的关系语言中为世袭财产列出和极端问题。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。