Mathematical Sciences: Model Theory, Geometry and Arithmetic

模型论，代数和几何项目的第一部分将继续发展p进和刚性亚解析集的理论。p进理论是由Denef和van den Dries开创的，刚性解析理论是由研究者提出的。这些发展是基于各种量词消去定理。本文还打算用这种量词消去的方法给出一些在实际情况下的定理的新的证明，并把这些新的证明转移到p进的和刚性的分析情况中。还考虑了与阿廷近似定理的联系的研究。项目的第2部分将继续研究整数和多项式环的加法和可除性语言中的各种丢番图问题。这将扩展研究者Denef和Pheidas早期的工作。看到关于代数对象（如幂级数环）的定理是如何通过从数学逻辑中提取的方法来严肃地解决的，这是令人着迷的，也就是说，这些方法主要处理表达定理的语言的特征。这并不是故事的全部，但看到一个人能以这种方式走多远，就能以一种非常有启发性的方式隔离其他障碍。

Part 1 of this project in model theory, algebra and geometry will continue the development of a theory of p-adic and rigid subanalytic sets. The p-adic theory was begun by Denef and van den Dries and the rigid analytic theory was developed by the investigator. These developments are based on various quantifier elimination theorems. It is also intended to give new proofs of theorems in the real case using this method of quantifier elimination and to transfer these new proofs also to the p-adic and rigid analytic cases. The investigation of connections with Artin Approximation Theorems is also contemplated. Section 2 of the project will continue the investigation of various diophantine problems in the language of addition and divisibility for the integers and polynomial rings. This would extend earlier work by the investigator, Denef and Pheidas. It is fascinating to see how theorems about algebraic objects such as rings of power series can be seriously addressed by methods drawn from mathematical logic, that is, methods which deal largely with features of the language in which the theorems are expressed. This is not the whole story, but seeing how far one can get in this way isolates other obstacles in a highly illuminating way.