Model theory has made substantial contributions to semialgebraic, subanalytic, p-adic, rigid and diophantine geometry. These applications range from a proof of the rationality of certain Poincare series associated to varieties over p-adic fields, to a proof of the Mordell-Lang conjecture for function fields in positive characteristic. In some cases (such as the latter) it is the most abstract aspects of model theory which are relevant. This book, originally published in 2000, arising from a series of introductory lectures for graduate students, provides the necessary background to understanding both the model theory and the mathematics behind these applications. The book is unique in that the whole spectrum of contemporary model theory (stability, simplicity, o-minimality and variations) is covered and diverse areas of geometry (algebraic, diophantine, real analytic, p-adic, and rigid) are introduced and discussed, all by leading experts in their fields.

• Introductory surveys for graduate students by top-notch contributors

### Contents

Overview Anand Pillay, Charles Steinhorn, and Deirdre Haskell; 1. Introduction to model theory David Marker; 2. Classical model theory of fields Lou van den Dries; 3. Model theory of differential fields David Marker; 4. A survey on the model theory of differential fields Zoé Chatzidakis; 5. Notes on o-minimality and variations Dugald Macpherson; 6. Stability theory and its variants Bradd Hart; 7. Subanalytical geometry Edward Bierstone and Pierre D. Milman; 8. Arithmetic and geometric applications of quantifier elimination for valued fields Jan Denef; 9. Abelian varieties and the Mordell-Lang conjecture Barry Mazur.