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·汉森将继续发展他的模型理论 Banach空间，并将研究非标准 分析，包括超有限集的描述集理论。 在 此外，他将研究逻辑的计算复杂性， 决策问题，包括真实的指数理论 领域 Carl Jockusch将研究经典递归理论， 与其他领域的联系。 特别是，他将研究 程度的不可解决性，无论是在其本身的权利和连接 与模型理论结构（特别是布尔代数和 某些群体）。 L.货车登德里斯将调查的可定义性 代数和解析结构。 特别是，他将 研究真实的序域展开式的O-极小性 与A有关的数字和问题。威尔基最近证明 这个有序域的展开的基本理论， 求幂是模型完成的。 他还将研究模型 局部场和Henselian场理论 数学逻辑是研究任何 以正式语言表达的纪律，这应该带来最 数学本身在它的关注之下。 代数与数论 而分析的某些部分是特别自然的主题， 这种待遇。 大部分的即时项目承担 办法的 调查人员最近工作的一个典型结果是 塔斯基提出的完整性猜想的证明：是否存在 结合多项式形成和 常数的取幂，从所有其他这样的真实 身份是可以推导出来的 （反例已知存在，如果 允许对变量取幂。） 另一个领域， 逻辑的自然作用是复杂性， 算法 调查人员巧妙地发现了这样的 利基以及在探索他们，以双方的明显利益 逻辑和其他相关学科。

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.