Mathematical Sciences: Model Theory, Geometry and Arithmetic

这个项目的第一部分将在模型理论、代数和几何中继续发展p-进和刚性次分析集理论。P-进理论是由Denef和van den Dries开创的，刚性分析理论是由研究者发展起来的。这些发展是基于各种量词消去定理的。用这种量词消去法给出了实际情况下定理的新证明，并将这些新证明也转移到p元和刚性分析情况。研究了与Artin逼近定理之间的联系。该项目的第二节将继续研究整数和多项式环的加法和整除语言中的各种丢番图问题。这将扩展调查员德内夫和菲达斯早先的工作。从数理逻辑中提取的方法，即主要处理表达定理的语言的特征的方法，可以认真地处理有关代数对象的定理，如幂级数的环，这是令人着迷的。这并不是故事的全部，但看到一个人通过这种方式能走多远，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.