Model Theoretic Classification Theory and Finite Combinatorics

模型理论是数学逻辑的一个分支，它试图理解驱动不同类型数学对象行为的共同结构现象。这一领域的一个关键思想是分界线的概念，最早是由希拉在20世纪70年代提出的。分界线可以被认为是某一类数学对象中的结构二分法。许多最重要的分界线对应于局部组合属性，这些属性对全局结构具有重要意义。在无限设置中，模型理论家使用分界线对示例进行分类并归纳其行为取得了巨大成功。然而，在有限情况下的扩展是有限的，很大程度上是由于关键的无穷大工具的失败。另一方面，极值和算术组合学是专注于有限设置的领域，但它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。这些领域发展了有限问题和工具，这些问题和工具对模型理论来说是新的，但它们与模型理论思想有着深刻的联系。该项目的目标是通过解决解决这些共同主题的极值和加性组合的有限问题来扩展有限设置下模型理论分界线的研究。更具体地说，该项目将侧重于寻找局部模型理论条件，这些条件对加法和极值组合定理的界和增长率具有强大的含义。这将在两个主要方向上实现。第一个将集中在加法组合学的问题上。这里的主要目标是识别素域上高维向量空间子集的结构二分类。例如，通过结构分解定理中改进的界来衡量，什么样的集合是最“驯服”的？“驯服”的集合可以用局部组合配置来表征吗？第二个方向将解决极值组合中的问题。具体来说，PI将继续研究有限关系语言中遗传性质的枚举和极值问题。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 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.