Bücher der Reihe Encyclopaedia of Mathematical Sciences

Projective duality is a very classical notion naturally arising in various areas of mathematics, such as algebraic and differential geometry, combinatorics, topology, analytical mechanics, and invariant theory, and the results in this field were until now scattered across the literature.

Wholeheartedly recommended to every student and user of mathematics, this is an extremely original and highly informative essay on algebra and its place in modern mathematics and science.

From the reviews: "These three bulky volumes [EMS 124, 125, 127] [...] provide an introduction to this rapidly developing theory. [...] These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of mathematics. Acta Scientiarum Mathematicarum

From the reviews of the first edition:"... Here ... a wealth of material is displayed for us, too much to even indicate in a review. ... Your reviewer was very impressed by the contents of both volumes (EMS 2 and 4), recommending them without any restriction." Mededelingen van het Wiskundig genootshap 1992

A brilliant and coherent summation of both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare.

This book presents a survey of the theory of general linear partial differential equations and, in particular, of equations with constant coefficients. It is written for researchers and graduate students in mathematics and theoretical physics.

Equivariant embeddings are essential tools in solving a variety of problems relating to homogenous spaces in linear algebraic groups. This volume classifies these embeddings using a 'combinatorial' data framework, with a special focus on spherical varieties.

Together with Theory of Operator Algebras I and III, this book presents the theory of von Neumann algebras and non-commutative integration focusing on the group of automorphisms and the structure analysis.

From the reviews of the first edition:"... Here ... a wealth of material is displayed for us, too much to even indicate in a review. ... Your reviewer was very impressed by the contents of both volumes (EMS 2 and 4), recommending them without any restriction." Mededelingen van het Wiskundig genootshap 1992

Throughout the history of invariant theory, computational methods have always been at the centre of attention. This book, the first volume of the subseries on "Invariant Theory and Algebraic Transformation Groups", provides a comprehensive overview of the algorithmic aspects of invariant theory.

The main topics covered include: constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, and Floer homology and gauge-theoretical invariants of homology cobordism.

In addition, it details some interesting applications of the general theory of vortices, such as applications in numerical methods, stability theory, and the theory of exact integration of equations of dynamics.

This book is a survey of the most important directions of research in transcendental number theory. For readers with no specific background in transcendental number theory, the book provides both an overview of the basic concepts and techniques and also a guide to the most important results and references.

This is the second volume of the new subseries "Invariant Theory and Algebraic Transformation Groups". Bialynicki-Birula is to present the main trends and achievements of research in the theory of quotients by actions of algebraic groups. This theory contains geometric invariant theory with various applications to problems of moduli theory.

This collection of surveys written by leading researchers of the fields - mathematicians, physicists or mathematical physicists - treat both mathematically rigourous results, and evolving physical theories where the methods are analytic or computational.

Thus, m the space of states (phase space) is a subset M of an Euclidean space M . For continuous time systems, the evolution rule may be a differential eq- tion: to each state x G M one associates the speed and direction in which the system is going to evolve from that state.

The book covers topics in the theory of algebraic transformation groups and algebraic varieties which are very much at the frontier of mathematical research.

The book covers topics in the theory of algebraic transformation groups and algebraic varieties which are very much at the frontier of mathematical research.

This EMS volume provides an exposition of the structure theory of Fano varieties, i.e. algebraic varieties with an ample anticanonical divisor. This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

This EMS volume gives an overview of the modern theory of elliptic boundary value problems, with contributions focusing on differential elliptic boundary problems and their spectral properties, elliptic pseudodifferential operators, and general differential elliptic boundary value problems in domains with singularities.

This book is a survey of the most important directions of research in transcendental number theory. For readers with no specific background in transcendental number theory, the book provides both an overview of the basic concepts and techniques and also a guide to the most important results and references.

This newly translated addition to the literature provides a comprehensive treatment of the fundamentals of geophysical hydrodynamics -as expressed in Dolzhansky's four decades of ground-breaking work at the Moscow Institute of Physics and Technology.

This book covers the modular invariant theory of finite groups. It details techniques for the computation of invariants for many modular representations of finite groups, especially the cyclic group of prime order, and includes many examples.

Contributions by three authors treat aspects of noncommutative geometry that are related to cyclic homology. The authors give rather complete accounts of cyclic theory from different points of view. Cyclic theory is the natural setting for a variety of general abstract index theorems.

[Gabriel and Roiter] are pioneers in this subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature.

From the reviews: "..., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject.

In sum, the volume under review is the first quarter of an important work that surveys an active branch of modern mathematics. it will be an invaluable reference and a companion to modern courses on several complex variables."

Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. Both articles give comprehensive bibliographies, so that it is possible to use this book as the starting point for a more detailed study of a particular topic of interest.