Model theory is a branch of mathematical logic that has found applications in several areas of algebra and geometry. It provides a unifying framework for the understanding of old results and more recently has led to significant new results, such as a proof of the Mordell-Lang conjecture for function fields in positive characteristic. Perhaps surprisingly, it is sometimes the most abstract aspects of model theory that are relevant to those applications. This book gives the necessary background for understanding both the model theory and the mathematics behind the applications. Aimed at graduate students and researchers, it contains introductory surveys by leading experts covering the whole spectrum of contemporary model theory (stability, simplicity, o-minimality and variations), and introducing and discussing the diverse areas of geometry (algebraic, diophantine, real analytic, p-adic, and rigid) to which the model theory is applied. The book begins with an introduction to model theory by David Marker. It then broadens into three components: pure model theory (Bradd Hart, Dugald Macpherson), geometry(Barry Mazur, Ed Bierstone and Pierre Milman, Jan Denef), and the model theory of fields (Marker, Lou van den Dries, Zoe Chatzidakis).
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.