Part 1 of this project in model theory, algebra and geometry will continue the development of a theory of p-adic and rigid subanalytic sets. The p-adic theory was begun by Denef and van den Dries and the rigid analytic theory was developed by the investigator. These developments are based on various quantifier elimination theorems. It is also intended to give new proofs of theorems in the real case using this method of quantifier elimination and to transfer these new proofs also to the p-adic and rigid analytic cases. The investigation of connections with Artin Approximation Theorems is also contemplated. Section 2 of the project will continue the investigation of various diophantine problems in the language of addition and divisibility for the integers and polynomial rings. This would extend earlier work by the investigator, Denef and Pheidas. It is fascinating to see how theorems about algebraic objects such as rings of power series can be seriously addressed by methods drawn from mathematical logic, that is, methods which deal largely with features of the language in which the theorems are expressed. This is not the whole story, but seeing how far one can get in this way isolates other obstacles in a highly illuminating way.
