Topics in Model Theory

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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).