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:
  • 金额:
    $ 16.43万
  • 依托单位:
  • 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

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.
模型理论是数理逻辑的一个分支,旨在理解驱动不同类型数学对象行为的常见结构现象。 该领域的一个重要思想是分界线的概念,由 Shelah 在 20 世纪 70 年代首次提出。分界线可以被认为是某一类数学对象内的结构二分法。 许多最重要的分界线对应于局部组合属性,这对全局结构具有重大影响。 在无限环境中,模型理论家使用分界线对示例进行分类并概括其行为取得了巨大成功。然而,有限设置的扩展受到限制,这主要是由于关键的无限工具的失败。 另一方面,极值组合学和算术组合学是专注于有限设置的领域,但它们研究许多与模型论相同的主题,例如局部结构与全局结构以及结构与随机性的相互作用。 这些领域开发了有限问题和工具,这些问题和工具对于模型理论来说是新的,但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极值和加性组合学的有限问题来扩展有限设置中模型理论分界线的研究,这些问题解决了这些共同的主题。更具体地说,该项目将重点寻找局部模型理论条件,这些条件对加性和极值组合学定理的界限和增长率具有强大的影响。 这将在两个主要方向上完成。 第一个将重点关注加法组合学的问题。 这里的主要目标是识别素数域上高维向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界限来衡量,什么类型的集合是最“驯服”的?“驯服”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学问题。 具体来说,PI 将继续致力于有限关系语言中遗传属性的枚举和极值问题。该奖项反映了 NSF 的法定使命,并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,被认为值得支持。

项目成果

期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
An improved bound for regular decompositionsof 3-uniform hypergraphs of bounded VC2-dimension
有界 VC2 维 3 一致超图正则分解的改进界
  • DOI:
    10.2140/mt.2023.2.325
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Terry, Caroline
  • 通讯作者:
    Terry, Caroline
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Caroline Terry其他文献

Quantitative structure of stable sets in finite abelian groups
有限阿贝尔群中稳定集的数量结构
A group version of stable regularity
稳定正则性的群版本

Caroline Terry的其他文献

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{{ truncateString('Caroline Terry', 18)}}的其他基金

CAREER: Model theoretic classification theory, Fourier analysis, and hypergraph regularity
职业:模型理论分类理论、傅立叶分析和超图正则性
  • 批准号:
    2239737
  • 财政年份:
    2023
  • 资助金额:
    $ 16.43万
  • 项目类别:
    Continuing Grant
Model Theoretic Classification Theory and Finite Combinatorics
模型理论分类理论和有限组合学
  • 批准号:
    1855711
  • 财政年份:
    2019
  • 资助金额:
    $ 16.43万
  • 项目类别:
    Standard Grant

相似海外基金

CAREER: Model theoretic classification theory, Fourier analysis, and hypergraph regularity
职业:模型理论分类理论、傅立叶分析和超图正则性
  • 批准号:
    2239737
  • 财政年份:
    2023
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    $ 16.43万
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    Continuing Grant
Model Theoretic Classification Theory and Finite Combinatorics
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模型理论分类、图组合学和拓扑动力学
  • 批准号:
    1600796
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    2016
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"Measure-theoretic dimension groups, classification of ergodic transformation, and topics in positivity"
“测度论维度组、遍历变换的分类以及积极性主题”
  • 批准号:
    3130-2012
  • 财政年份:
    2015
  • 资助金额:
    $ 16.43万
  • 项目类别:
    Discovery Grants Program - Individual
"Measure-theoretic dimension groups, classification of ergodic transformation, and topics in positivity"
“测度论维度组、遍历变换的分类以及积极性主题”
  • 批准号:
    3130-2012
  • 财政年份:
    2014
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    Discovery Grants Program - Individual
"Measure-theoretic dimension groups, classification of ergodic transformation, and topics in positivity"
“测度论维度组、遍历变换的分类以及积极性主题”
  • 批准号:
    3130-2012
  • 财政年份:
    2013
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    $ 16.43万
  • 项目类别:
    Discovery Grants Program - Individual
"Measure-theoretic dimension groups, classification of ergodic transformation, and topics in positivity"
“测度论维度组、遍历变换的分类以及积极性主题”
  • 批准号:
    3130-2012
  • 财政年份:
    2012
  • 资助金额:
    $ 16.43万
  • 项目类别:
    Discovery Grants Program - Individual
Classification and representation theoretic study of the functional equation-spaces
函数方程空间的分类与表示理论研究
  • 批准号:
    20540021
  • 财政年份:
    2008
  • 资助金额:
    $ 16.43万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Index theoretic approaches to the classification of positive scalar curvature
正标量曲率分类的索引理论方法
  • 批准号:
    5453910
  • 财政年份:
    2005
  • 资助金额:
    $ 16.43万
  • 项目类别:
    Priority Programmes
Index theoretic approaches to the classification of positive scalar curvature
正标量曲率分类的索引理论方法
  • 批准号:
    5406956
  • 财政年份:
    2003
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