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.沃德·亨森将研究 指数函数，并将应用非标准分析， 从模型论到Banach空间的研究， 分析的其他主题。 此外，他还将试图 确定各种特定的计算复杂度 一阶理论，并描述那些理论， 可以在多项式空间中计算。 他还将调查 高度均匀的结构和它们的 自同构 Carl Jockusch将学习经典递归 理论，特别是将研究的程度 对角非递归函数的相对不可解性 递归可枚举性和递归可枚举度 相对于某些允许的参数， 执行。 卢·货车·登·德里斯将学习代数决策 问题，如交换环的字问题和 求A的单位群的生成元问题 生成交换环 他也将继续他的 研究代数解析结构的可定义性。 一些调查将完善的技术， 数学逻辑，有些人会把它们应用到经常出现的问题上， 被视为理论计算机科学或代数。 Jockusch的工作将是前一种，而一部分 亨森和货车登德里斯的工作将是后者。 为 例如，Henson将继续评估 不同的数学理论，并将寻求方法， 容易识别复杂度较低的那些。

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.