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This book provides an entry into some the key areas of research in contemporary model theory. Model theory, a branch of mathematical logic, is an exciting and vibrant discipline. Advances in pure model theory drive applications in algebra, algebraic geometry, analysis, combinatorics and number theory. The contributing authors are leaders in the field, both senior and junior, including Anand Pillay, Zoé Chatzidakis, Gabriel Conant, Caroline Terry, Itay Kaplan, Rahim Moosa and Silvain Rideau-Kikuchi.
This book introduces readers to contemporary stability theory, the model theory of finite and pseudo-finite fields, the model theory of differential fields, and the basics of simplicity theory and NSOP1 theories, which culminate in proving the symmetry of Kim-independence. Contributors give a detailed proof of a qualitative version of the Malliaris-Shelah regularity lemma for stable graphs using only basic local stability theory and an ultraproduct construction. Additionally, contributors give self-contained exposition of two cornerstones of the geometric theory of algebraically closed valued fields. The first is a description of the definable sets in the guise of an elimination of quantifiers, essentially dating back to Robinson's work. The second is a description of all interpretable set in the guise of the Haskell-Hrushovski-Macpherson elimination of imaginaries.
Table des matières
Preface.- Introduction, Model Theory, and Stability.- Notes on the Model Theory of Finite and Pseudo-Finite Fields.- Pseudo-Finite Proofs of the Stable Graph Regularity Lemma.- Notes on the Basics of Simple Theories and NSOP_1.- Six Lectures on Model Theory and Differentia-Algebraic Geometry.- Model Theory of Valued Fields.
