Mathematical Sciences: Model Theory and Analysis

DMS 9626628 C.Ward Henson C.Ward Henson寻求使用逻辑中的方法来解决数学和计算机科学的其他领域中的问题，并将逻辑用作形成新概念和在其中提出新问题的基础。这个项目有以下主要方面：(A)Henson逻辑的进一步发展，用于基于Banach空间(或更一般的，度量空间)的模型，并继续将该理论应用于泛函分析问题；(B)从模型论的观点研究指数运算和其他初等函数；这项研究的目标包括确定实指数域的一阶理论的计算复杂性，并公理由诸如Gamma函数等其他初等函数满足的恒等式；(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，来自逻辑的技术不仅澄清了数学的基础，而且为解决数学问题提供了新的工具。这些工具在那些接近代数的数学领域特别有效。这个项目专注于使这些相同的逻辑工具在更接近拓扑学和几何学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.