Deirdre Haskell – författare

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.

This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.

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.

This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.

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.