Topics in Model Theory

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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.