Topics in Model Theory

项目编号：dms -0300639首席研究员：Anand pillay该提案涉及模型论（数理逻辑的一个分支）及其应用的几个领域的问题。另一方面，涉及简单理论模型的可定义性和精细结构的问题，将用模型的初等对理论来研究。同时，对附属于任何一阶理论的伽罗瓦群——拉斯卡群的结构问题，提出了一阶逻辑和描述集合论之间新的猜想关系。应用方面主要涉及微分场的射流空间方法、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是模型理论预测进一步的可定义集合来自代数几何的方式，其结构越刚性，以及在研究中的具体情况下这种行为的数学含义。这个建议属于数学逻辑的一个分支——模型论。现代数理逻辑发展于二十世纪。虽然目前的建议没有直接涉及可计算性问题，但值得注意的是，像图灵这样的数理逻辑学家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，模型理论关注的是对数学家用来表达自己和定义数学对象的语言的详细研究。像罗宾逊和塔斯基这样的模型理论家表明，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。非标准分析的发展就是一个例子。在过去的15年里，模型理论在数论和几何等数学领域的应用取得了惊人的进展。除其他事项外，提案人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.