Principal Investigator: Anand Pillay
The proposal covers problems in several areas of model theory (a branch of mathematical logic) and its applications. On the pure side are problems involving definability and fine structure in models of simple theories, which will be approached using the theory of elementary pairs of models. Also questions on the structure of the Lascar group, a certain Galois group attached to any first order theory, raise new conjectural relationships between first order logic and descriptive set theory. The applied side of the proposal deals mainly with jet-space methods in differential fields, the Galois theory of differential equations in positive characteristic, and the model theory of Kahler manifolds. Among the common themes on the applied side is the manner in which model theory predicts that the further a definable set is from algebraic geometry, the more rigid is its structure, and the mathematical implications of this behaviour in the concrete situations under investigation.
The proposal is in a certain branch of mathematical logic called model theory. Modern mathematical logic developed in the twentieth century. Although the present proposal does not touch directly on computability questions, it is worth remarking that mathematical logicians such as Turing provided the first rigorous notions of computability and played a role in the development of actual computers. Model theory, on the other hand, is concerned with a detailed study of the language in which mathematicians express themselves and define mathematical objects. Model-theorists such as Robinson and Tarski showed how such preoccupations can have wide-ranging implications for mathematics itself, and so indirectly for science as a whole. The development of nonstandard analysis provides an example. The past fifteen years have seen striking advances in applications of model theory to areas of mathematics such as number theory and geometry. The proposer plans, among other things, to extend these advances, using new methods he has recently developed.
