Topics in Model Theory

AbstractAward：DMS-0300639首席研究员：Anand Pillay该提案涵盖了模型理论（数理逻辑的分支）及其应用的几个领域的问题。纯粹的问题涉及简单理论模型的可定义性和精细结构，这将使用基本模型对理论来处理。也问题的结构，拉斯卡组，一定的伽罗瓦集团重视任何一阶理论，提出新的理论关系之间的一阶逻辑和描述集理论。该建议的应用方面主要涉及微分场的喷流空间方法、正特征微分方程的伽罗瓦理论和Kahler流形的模型理论。在应用方面的共同主题之一是模型论预测的方式，即一个可定义的集合离代数几何越远，它的结构就越严格，以及在所研究的具体情况下这种行为的数学含义。现代数理逻辑发展于二十世纪。虽然目前的建议没有直接触及可计算性问题，但值得注意的是，数学逻辑学家如图灵提供了第一个严格的可计算性概念，并在实际计算机的发展中发挥了作用。另一方面，模型论关注的是对数学家表达自己和定义抽象对象的语言的详细研究。模型理论家如罗宾逊和塔斯基表明了这种关注如何对数学本身产生广泛的影响，从而间接地对整个科学产生影响。非标准分析的发展提供了一个范例。在过去的15年里，模型论在数学领域的应用取得了惊人的进展，如数论和几何。除此之外，提议者计划使用他最近开发的新方法来扩展这些进步。

AbstractAward: DMS-0300639Principal Investigator: Anand PillayThe proposal covers problems in several areas of model theory (abranch of mathematical logic) and its applications. On the pureside are problems involving definability and fine structure inmodels of simple theories, which will be approached using thetheory of elementary pairs of models. Also questions on thestructure of the Lascar group, a certain Galois group attached toany first order theory, raise new conjectural relationshipsbetween first order logic and descriptive set theory. The appliedside of the proposal deals mainly with jet-space methods indifferential fields, the Galois theory of differential equationsin positive characteristic, and the model theory of Kahlermanifolds. Among the common themes on the applied side is themanner in which model theory predicts that the further adefinable set is from algebraic geometry, the more rigid is itsstructure, and the mathematical implications of this behaviour inthe concrete situations under investigation.The proposal is in a certain branch of mathematical logic calledmodel theory. Modern mathematical logic developed in thetwentieth century. Although the present proposal does not touchdirectly on computability questions, it is worth remarking thatmathematical logicians such as Turing provided the first rigorousnotions of computability and played a role in the development ofactual computers. Model theory, on the other hand, is concernedwith a detailed study of the language in which mathematiciansexpress themselves and define mathematicalobjects. Model-theorists such as Robinson and Tarski showed howsuch preoccupations can have wide-ranging implications formathematics itself, and so indirectly for science as a whole. Thedevelopment of nonstandard analysis provides an example. The pastfifteen years have seen striking advances in applications ofmodel theory to areas of mathematics such as number theory andgeometry. The proposer plans, among other things, to extend theseadvances, using new methods he has recently developed.