Model Theory and Analysis

模型理论与分析

• 批准号: 1101316
• 负责人: Isaac Goldbring
• 金额: $ 9.59万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2012-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101316&HistoricalAwards=false
• 关键词: Model; Theory; Analysis

Goldbring proposes to continue the use of techniques from logic, specifically model theory, to answer questions about structures in analysis. More specifically, he plans on: (1) using the techniques of nonstandard analysis to investigate problems in infinite-dimensional Lie theory and geometric group theory; and (2) continuing the development of continuous logic and model theory for metric structures. In regards to item (1), he plans on investigating how one can use the technique of hyperfinite approximation in infinite-dimensional Lie theory by embedding infinite-dimensional objects inside of hyperfinite objects in the hopes that the finite-dimensional theory will apply to the hyperfinite objects and the infinite-dimensional objects embedded in them. In regards to (2), Goldbring plans on continuing the study of foundational aspects of continuous logic, including definability of sets and functions and the behavior of independence relations in various metric structures.Model theory, a branch of mathematical logic, studies what properties of mathematical structures can be described in terms of first-order logic. This understanding can often lead to solutions to problems in various mathematical areas by taking general model-theoretic facts and interpreting them in specific contexts. This approach has been most successful in discrete situations, such as algebra and combinatorics. Recently, a model theory has been developed for structures arising in analysis, namely those for which a natural notion of distance exists. This model theory could then prove useful for solving problems in areas where classical model theory has yet to be able to help. That being said, model theory, in the form of nonstandard analysis, has found applications in analysis by providing a rigorous notion of infinitesimal and infinite numbers. Nonstandard analysis has found applications in such diverse areas as number theory, probability theory, and economics. This project proposes to use nonstandard analysis to answer questions in Lie theory, which is a very important mathematical area used in physics and engineering.

戈德布林建议继续使用逻辑技术，特别是模型理论，来回答有关分析结构的问题。更具体地说，他计划：(1)利用非标准分析技术研究无穷维李论和几何群论中的问题；(2)继续发展度量结构的连续逻辑和模型理论。关于第(1)项，他计划研究如何利用无限维李氏理论中的超有限逼近技术，将无限维对象嵌入到超有限对象中，希望有限维理论能够应用到超有限对象和嵌入其中的无限维对象中。关于(2)，Goldbring计划继续研究连续逻辑的基本方面，包括集合和函数的可定义性以及各种度量结构中独立关系的行为。模型论是数理逻辑的一个分支，研究数学结构的哪些性质可以用一阶逻辑来描述。这种理解通常可以通过采用一般的模型理论事实并在特定的上下文中解释它们来解决各种数学领域的问题。这种方法在离散的情况下最为成功，例如代数和组合学。最近，对于在分析中产生的结构，即那些存在自然距离概念的结构，已经发展了一种模型理论。然后，这个模型理论可以证明对解决经典模型理论尚未能够帮助的领域的问题是有用的。也就是说，模型理论，以非标准分析的形式，通过提供无限小和无限数的严格概念，已经在分析中找到了应用。非标准分析在数论、概率论和经济学等不同领域都有应用。李理论是物理和工程中一个非常重要的数学领域，本项目提出用非标准分析来回答李理论中的问题。