Mathematical Sciences: Model Theory and Analysis

C. Ward Henson致力于使用逻辑方法来解决数学和计算机科学其他领域的问题，并将逻辑作为形成新概念和提出新问题的基础。该项目有以下几个主要方面：(a)进一步发展Henson基于Banach空间（或者更一般地说，基于度量空间）的模型逻辑，并继续将该理论应用于泛函分析问题；(b)从模型论的角度研究指数和其他初等函数；本研究的目的包括确定实指数场一阶理论的计算复杂度，并公理化其他初等函数如γ函数所满足的恒等式；(c)继续研究逻辑决策问题的计算复杂性，特别是那些来自数学最核心领域的问题。这个项目强调符号逻辑与数学和计算机科学的其他领域之间的关系。在过去的40年里，逻辑技术不仅阐明了数学的基础，而且为解决数学问题提供了新的工具。这些工具在接近代数的数学领域特别有效。这个项目的重点是使这些相同的逻辑工具在更接近拓扑和几何的数学领域同样有效。

DMS 9626628 C. Ward Henson C. Ward Henson seeks to use methods from logic to solve problems within other areas of mathematics and computer science, and to use logic as a basis for formulating new concepts and posing new problems within them. This project has the following main aspects: (a) the further development of Henson's logic for models based on Banach spaces (or, more generally, on metric spaces), and the continued application of this theory to problems of functional analysis; (b) the study of exponentiation and other elementary functions from the model-theoretic point of view; objectives of this study include determining the computational complexity of the first order theory of the real exponential field and axiomatizing the identities satisfied by such other elementary functions as the Gamma function; (c) continued investigation of the computational complexity of logical decision problems, especially those arising from the most central areas of mathematics. This project emphasizes the relations between symbolic logic and other areas of mathematics and computer science. During the past 40 years techniques from logic have not only clarified the foundations of mathematics, but have contributed new tools for solving mathematical problems. These tools have been particularly effective in those areas of mathematics close to algebra. This project concentrates on making those same logic tools equally effective in the areas of mathematics closer to topology and geometry.