Model Theory, Algebra, and Geometry

by Edited by Deirdre Haskell , Anand Pillay , Charles Steinhorn

• ISBN13: 9780521143493
• ISBN10: 0521143497
• Format: Paperback
• Copyright: 2010-06-10
• Publisher: Cambridge University Press

Summary

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).

Table of Contents

Overview

Introduction to model theory

Classical model theory of fields

Model theory of differential fields

A survey on the model theory of differential fields

Notes on o-minimality and variations

Stability theory and its variants

Subanalytical geometry Edward Bierstone

Arithmetic and geometric applications of quantifier elimination for valued fields

Abelian varieties and the Mordell-Lang conjecture

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