Böcker i Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics-serien

This book uses algebraic tools to study the elementary properties of classes of fields and related algorithmic problems. The first part covers foundational material on infinite Galois theory, profinite groups, algebraic function fields in one variable and plane curves. It provides complete and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, together with material on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability and nonstandard model theory, including a nonstandard proof of Hilbert's irreducibility theorem. The focus then turns to the study of pseudo algebraically closed (PAC) fields, related structures and associated decidability and undecidability results. PAC fields (fields K with the property that every absolutely irreducible variety over K has a rational point) first arose in the elementary theory of finite fields and have deep connections with number theory.Thisfourth edition substantially extends, updates and clarifies the previous editions of this celebrated book, and includes a new chapter on Hilbertian subfields of Galois extensions. Almost every chapter concludes with a set of exercises and bibliographical notes. An appendix presents a selection of open research problems. Drawing from a wide literature at the interface of logic and arithmetic, this detailed and self-contained text can serve both as a textbook for graduate courses and as an invaluable reference for seasoned researchers.

This book pioneers a nonlinear Fredholm theory in a general class of spaces called polyfolds.

by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra.

Two volume work containing a contemporary account on "Positivity in Algebraic Geometry". 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".A good deal of the material has not previously appeared in book form. Volume II is more at the research level and somewhat more specialized than Volume I.

This monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry.

Some years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjec tures. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils.

This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field.

This self-contained introduction treats thin geometries and thick buildings from a diagrammatic perspective, covers polar geometries whose projective planes are Desarguesian and offers a basic reference for study of diagram geometry. Includes many examples.

This book covers the basics of Clifford algebras and spinor modules, with applications to the theory of Lie groups. Topics include Petracci's proof of the Poincare-Birkhoff-Witt theorem, quantized Weil algebras, Duflo's theorem for quadratic Lie algebras and more.

Neron models were invented by A. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Neron models from the point of view of Grothendieck's algebraic geometry.

The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem.

Offering the first comprehensive treatment of this subject in book form, this volume presents the elements of a general theory for flows on three-dimensional compact boundaryless manifolds, encompassing flows with equilibria accumulated by regular orbits.

This two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity.

This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.

This book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur.Its connecting thread is a question about lattices of rank 24: the problem of p-neighborhoods between Niemeier lattices. This question, whose expression is quite elementary, is in fact very natural from the automorphic point of view, and turns out to be surprisingly intriguing. We explain how the new advances in the Langlands program mentioned above pave the way for a solution. This study proves to be very rich, leading us to classical themes such as theta series, Siegel modular forms, the triality principle, L-functions and congruences between Galois representations.This monograph is intended for any mathematician with an interest in Euclidean lattices, automorphic forms or number theory. A large part of it is meant to be accessible to non-specialists.

This updated book serves both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. This revised edition contains new results, improved proofs, typographical corrections, and an enlarged bibliography.

This, the fourth edition of Stuwe's book on the calculus of variations, surveys new developments in this exciting field. Recently discovered results for backward bubbling in the heat flow for harmonic maps or surfaces are discussed.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordered Fields, Real Closed Fields . . . . . . 1 Ordered Fields, Real Fields . " . Semi-algebraic Sets . Curve-selection Lemma .

The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics.

In the 19 years which passed since the first edition was published, several important developments have taken place in the theory of surfaces.

This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject.The author begins by carefully developing relevant notions in topology, profinite groups and homology, including free products of profinite groups, cohomological methods in profinite groups, and fixed points of automorphisms of free pro-p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups, with sections on finitely generated subgroups of free groups, separability conditions in free and amalgamated products, and algorithms in free groups and finite monoids.Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further reading.

The theme of this book is an examination of the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds, offering the first complete account of Oka-Grauert theory and its modern extensions.

This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori's abelian category of mixed motives.

This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. The author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature.

This second volume of Analysis in Banach Spaces, Probabilistic Methods and Operator Theory, is the successor to Volume I, Martingales and Littlewood-Paley Theory.

The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory.