Topics in Model Theory

皮莱计划对稳定和简单的理论，特别是超简单场的结构，微分场及其伽罗瓦群，以及0 -极小结构上的群进行研究。简单理论是一类一阶理论，既包括稳定理论，也包括典型的同型结构理论。他们的模型支持一种独立的概念。它专门研究代数闭域的代数自由。简单理论的研究提出了模型论的一些新思想，包括可定义性的广义概念和伽罗瓦理论。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简单域的例子。皮莱将研究反向问题：一个超简单场是否伪代数封闭，即F上的每一个不可约的变量是否都有一个F有理点。微分环和微分场是作为微分方程代数研究的一部分而发展起来的对象。利用稳定性理论的工具对这类环和场的模型理论研究，在微分代数几何和丢番图几何中都产生了影响。Pillay将研究微分伽罗瓦理论的逆问题，以及某些代数微分方程（painlevel的那些）解的精细结构。在0极小结构中可定义的群的范畴类似于实李群的范畴。在半简单的情况下，这种关系非常紧密。组。Pillay将尝试将其扩展到更广泛的群（代数群的阿贝尔、可定义紧、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).