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将继续发展他的Banach空间模型理论，并将研究非标准分析的主题，包括超有限集的描述集理论。此外，他将研究逻辑决策问题的计算复杂性，包括实指数场理论。Carl Jockusch将研究经典递归理论及其与其他领域的联系。特别是，他将研究不可解的程度，无论是在他们自己的权利和连接的模型理论结构（特别是布尔代数和某些群）。L. van den Dries将研究代数和解析结构的可定义性。特别是，他将研究实数有序域展开中的o -极小性，以及与A. Wilkie最近证明该有序域的幂展开式的基本理论是模型完备的相关问题。他还将研究局部场和亨塞利亚场的模型理论。数学逻辑是研究以形式语言表达的任何学科的天然工具，它应该把大多数数学本身置于其重点之下。代数和数论以及分析的某些部分是这种处理的特别自然的主题。即时项目的大部分内容都证明了这一点。研究人员最近工作的一个典型结果是证明了Tarski提出的一个完备性猜想：是否存在一个有限的真恒等式，它结合了多项式的形成和常数的幂，从而可以推导出所有其他的真恒等式？（如果允许对变量取幂，则已知存在反例。）逻辑可以发挥自然作用的另一个领域是算法的复杂性。研究者们在寻找和探索这些利基方面很有独创性，这对逻辑学和其他相关学科都有明显的好处。

The project will concern mathematical logic and its connections with other areas of mathematics, especially algebra and analysis. C. Ward Henson will continue developing his model theory for Banach spaces and will investigate topics in nonstandard analysis, including descriptive set theory on hyperfinite sets. In addition, he will study the computational complexity of logical decision problems, including the theory of the real exponential field. Carl Jockusch will work in classical recursion theory and its connections with other areas. In particular, he will study degrees of unsolvability, both in their own right and in connection with model-theoretic structures (especially Boolean algebras and certain groups). L. van den Dries will investigate definability in algebraic and analytic structures. In particular, he will investigate O-minimality in expansions of the ordered field of real numbers and questions connected with A. Wilkie's recent proof that the elementary theory of the expansions of this ordered field by exponentiation is model complete. He will also study the model theory of local and Henselian fields. Mathematical logic is a natural tool for studying any discipline expressed in a formal language, which should bring most of mathematics itself under its focus. Algebra and number theory and certain parts of analysis are particularly natural subjects for this type of treatment. Large parts of the instant project bear this out. A typical result of the investigators' recent work is the proof of a completeness conjecture raised by Tarski: Is there a finite list of true identities combining polynomial formation and exponentiation of constants from which all other such true identities can be derived? (Counterexamples are known to exist if exponentiation of variables is allowed.) Another area in which a natural role for logic can be found is the complexity of algorithms. The investigators have been ingenious in finding such niches as well as in exploring them, to the obvious benefit of both logic and the other discipline involved.