Model Theory, Algebra, and Geometry (Mathematical Sciences Research Institute Publications, Series Number 39)

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). Model theory has, in recent years, found substantial applications in semialgebraic, subanalytic, p-adic, rigid and diophantine geometry. This authoritative volume provides the necessary background to understanding both model theory and the mathematics underlying its applications. The book is unique in that the whole spectrum of contemporary model theory is covered and diverse areas of geometry are introduced and discussed, all by leading experts in their fields. The book begins with an introduction to model theory by David Marker, and then broadens into coverage of pure model theory, geometry, and the model theory of fields. Model theory has, in recent years, made substantial contributions to semialgebraic, subanalytic, p-adic, rigid and diophantine geometry. This book provides the necessary background to understanding the theory and mathematics In model theory we associate to a structure M invariants of a logical nature like Th(M), the set of first-order sentences which are true in M.