Topics in Model Theory

Pillay计划开展稳定和简单理论的研究，特别是超单场的结构研究，微分场及其Galois群的研究，以及o-极小结构上的群的研究。简单理论是一类一阶理论，既包括稳定理论，也包括典型的异类结构理论。他们的模型支持碘独立的概念。它专门研究在代数闭域的情况下的IO代数Freenessli。对简单理论的研究在模型理论中提出了一些新的思想，包括广义可定义性概念和伽罗瓦理论。皮莱计划研究一般理论和精细结构理论中的几个悬而未决的问题。具有小伽罗瓦群的伪代数闭域是超单域的例子。Pillay将研究相反的问题：是超单域伪代数闭的吗，即F上的每个不可约簇是否都有F-有理点。微环和场是作为微分方程组的代数研究的一部分而发展起来的对象。这种环和场的模型理论研究，利用稳定性理论的工具，在微分代数几何和丢番图几何中都产生了影响。Pillay将研究关于微分Galois理论的反问题，以及某些代数微分方程解的精细结构(Painleve的那些)。可在o-极小结构中定义的群的范畴类似于实李群的范畴。在ifSemiimplel的情况下，这种关系非常紧密。组。Pillay将试图将其扩展到更广泛的群类别(代数群的交换、可定义紧子群、可定义子群)。

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).