Model Theory and Analysis

戈德布赖恩建议继续使用逻辑中的技术，特别是模型理论，来回答分析中关于结构的问题。更具体地说，他计划：(1)使用非标准分析技术来研究无限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，他计划研究如何通过将无限维对象嵌入到超有限对象中来使用无限维李理论中的超有限逼近技术，希望有限维理论将应用于超有限对象和嵌入其中的无限维对象。关于(2)，戈德布赖恩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。模型理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获取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不同数学领域的问题。这种方法在离散情况下最为成功，例如代数和组合学。最近，对于分析中出现的结构，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，这种模型理论可能会被证明对解决经典模型理论尚未能够提供帮助的领域的问题很有用。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穷大的严格概念，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得到了广泛的应用。本项目提出使用非标准分析来回答李理论中的问题，这是一个在物理和工程中使用的非常重要的数学领域。

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.