Mathematical Sciences: Logic and Analysis

该项目将涉及数学逻辑及其与其他数学领域的联系，特别是代数和分析。C. Ward Henson将研究指数函数的模型论，并将应用非标准分析和模型论中的技术来研究巴拿赫空间和分析中的其他主题。此外，他将尝试确定各种特定一阶理论的计算复杂度，并描述那些在多项式空间中可计算的理论。他还将研究高度同构的结构及其自同构群。Carl Jockusch将研究经典递归理论，特别是研究对角非递归函数的不可解度，相对递归可枚举度，以及相对于某些允许参数执行的递归可枚举度。Lou Van den Dries将研究代数决策问题，如交换环的字问题和寻找有限生成交换环的单元群的生成子问题。他还将继续研究代数解析结构的可定义性。其中一些研究将改进数学逻辑的技术，还有一些将把它们应用于通常被视为理论计算机科学或代数的问题。Jockusch的作品将属于前者，而Henson和van den Dries的部分作品将属于后者。例如，汉森将继续评估不同数学理论的相对复杂性，并将寻找方法来识别复杂性较低的理论。

The project will concern mathematical logic and its connections with other areas of mathematics, especially algebra and analysis. C. Ward Henson will study the model theory of the exponental function and will apply nonstandard analysis and techniques from model theory to the study of Banach spaces and other topics from analysis. In addition, he will attempt to determine the computational complexity of various specific first-order theories and to characterize those theories which are computable in polynomial space. He will also investigate highly homogeneous structures and their groups of automorphisms. Carl Jockusch will work in classical recursion theory and in particular will study the degrees of unsolvability of diagonally nonrecursive functions, relative recursive enumerability, and recursively enumerable degrees relative to which certain permitting arguments may be performed. Lou Van den Dries will study algebraic decision problems such as the word problem for commutative rings and the problem of finding generators for the group of units of a finitely generated commutative ring. He also will continue his work on definability in algebraic-analytic structures. Some of the investigations will refine the techniques of mathematical logic, and some will apply them to questions often viewed as theoretical computer science or as algebra. Jockusch's work will be of the former kind, while a portion of Henson's and of van den Dries' work will be of the latter. For example, Henson will continue to assess the relative complexity of different mathematical theories and will seek ways to recognize easily ones with lower measures of complexity.