Topics in Model Theory

皮莱计划开展稳定和简单理论的研究，特别是超简单域的结构，微分域及其伽罗瓦群，以及o-极小结构上的群。简单理论是一类一阶理论，既包括稳定理论，也包括典型的ifgeneric结构理论。他们的模型支持一个独立于事物的概念。它专门研究代数闭域中的ioalgebraically freenessli。对简单理论的研究在模型论中提出了一些新的观点，包括广义的可定义性概念和伽罗瓦理论。皮莱计划调查几个开放的问题，在一般理论和精细结构理论。具有小伽罗瓦群的伪代数闭域是超单域的例子。皮莱将调查的反向问题：是一个超简单的领域pseudoalgebraically封闭，即每一个不可约品种超过F有一个F-合理的点。微分环和领域的对象开发的一部分，代数研究微分方程。使用稳定性理论的工具，对这种环和域的模型理论研究，对微分代数几何和丢番图几何产生了影响。皮莱将调查逆问题相对于微分伽罗瓦理论，以及精细结构的解决方案的某些代数微分方程（那些Painleve）。类别的群体定义在一个O-最小的结构类似于类别的真实的李群。在ifsemisimplel的情况下，这种关系非常紧密。组皮莱将试图扩大这一更广泛的类的群体（阿贝尔，可定义的紧凑，可定义的子群的代数群）。

Pillay plans to carry out research on stable and simple theories, especially the structure of supersimple fields, on differential fields and their Galois groups, and on groups over o- minimal structures.Simple theories are a class of first order theories, including both stable theories and typically the theories of ifgeneric structuresll. Their models support a notion of ioindependencel. which specialises to ioalgebraic freenessli in the case of algebraically closed fields. The study of simple theories has thrown up some new ideas in model theory involving generalized notions of definability and Galois theory. Pillay plans to investigate several open questions in both the general theory and fine structure theory. Pseudo algebraically closed fields with small Galois group are examples of supersimple fields. Pillay will investigate the reverse problem: is a supersimple field pseudoalgebraically closed, namely does every irreducible variety over F have an F- rational point.Differential rings and fields are objects developed as part of the algebraic study of differential equations. The model theoretic study of such rings and fields, using tools of stability theory, has had an impact in differential algebraic geometry as well as diophantine geometry. Pillay will investigate inverse problems with respect to differential Galois theory, as well as the fine structure of solutions of certain algebraic differential equations (those of Painleve).The category of groups definable in an o-minimal structure resembles the category of real Lie groups. The relationship is very tight in the case of ifsemisimplel. groups. Pillay will try to extend this to wider classes of groups (abelian, definably compact, definable subgroups of algebraic groups).