This book contains two contributions: "Combinatorial and Asymptotic Methods in Algebra" by V.A. Ufnarovskij is a survey of various combinatorial methods in infinite-dimensional algebras, widely interpreted to contain homological algebra and vigorously developing computer algebra, and narrowly interpreted as the study of algebraic objects defined by generators and their relations. The author shows how objects like words, graphs and automata provide valuable information in asymptotic studies. The main methods emply the notions of Grobner bases, generating functions, growth and those of homological algebra. Treated are also problems of relationships between different series, such as Hilbert, Poincare and Poincare-Betti series. Hyperbolic and quantum groups are also discussed. The reader does not need much of background material for he can find definitions and simple properties of the defined notions introduced along the way. "Non-Associative Structures" by E.N.Kuz'min and I.P.Shestakov surveys the modern state of the theory of non-associative structures that are nearly associative. Jordan, alternative, Malcev, and quasigroup algebras are discussed as well as applications of these structures in various areas of mathematics and primarily their relationship with the associative algebras. Quasigroups and loops are treated too. The survey is self-contained and complete with references to proofs in the literature. The book will be of great interest to graduate students and researchers in mathematics, computer science and theoretical physics."

This two-part EMS volume provides a succinct summary of complex algebraic geometry, coupled with a lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties. An excellent companion to the older classics on the subject.

Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

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 book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.

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.

Group theory is one of the most fundamental branches of mathematics. This volume of the Encyclopaedia is devoted to two important subjects within group theory. The first part of the book is concerned with infinite groups. The authors deal with combinatorial group theory, free constructions through group actions on trees, algorithmic problems, periodic groups and the Burnside problem, and the structure theory for Abelian, soluble and nilpotent groups. They have included the very latest developments; however, the material is accessible to readers familiar with the basic concepts of algebra. The second part treats the theory of linear groups. It is a genuinely encyclopaedic survey written for non-specialists. The topics covered includethe classical groups, algebraic groups, topological methods, conjugacy theorems, and finite linear groups. This book will be very useful to allmathematicians, physicists and other scientists including graduate students who use group theory in their work.

The first contribution by Carter covers the theory of finite groups of Lie type, an important field of current mathematical research. In the second part, Platonov and Yanchevskii survey the structure of finite-dimensional division algebras, including an account of reduced K-theory.

From the reviews: ..". [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. ..." --The Mathematical Gazette

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

From the reviews: ..". The book under review consists of two monographs on geometric aspects of group theory ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. 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. ..." Bulletin of the London Mathematical Society, 1996

This two-part volume contains numerous examples and insights on various topics. The authors have taken pains to present the material rigorously and coherently. This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetic algebraic geometry, complex analysis and related fields.

This book contains two contributions: "Combinatorial and Asymptotic Methods in Algebra" by V.A. Ufnarovskij is a survey of various combinatorial methods in infinite-dimensional algebras, widely interpreted to contain homological algebra and vigorously developing computer algebra, and narrowly interpreted as the study of algebraic objects defined by generators and their relations. The author shows how objects like words, graphs and automata provide valuable information in asymptotic studies. The main methods emply the notions of Grobner bases, generating functions, growth and those of homological algebra. Treated are also problems of relationships between different series, such as Hilbert, Poincare and Poincare-Betti series. Hyperbolic and quantum groups are also discussed. The reader does not need much of background material for he can find definitions and simple properties of the defined notions introduced along the way. "Non-Associative Structures" by E.N.Kuz'min and I.P.Shestakov surveys the modern state of the theory of non-associative structures that are nearly associative. Jordan, alternative, Malcev, and quasigroup algebras are discussed as well as applications of these structures in various areas of mathematics and primarily their relationship with the associative algebras. Quasigroups and loops are treated too. The survey is self-contained and complete with references to proofs in the literature. The book will be of great interest to graduate students and researchers in mathematics, computer science and theoretical physics."

The algebra of square matrices of size n ~ 2 over the field of complex numbers is, evidently, the best-known example of a non-commutative alge 1 bra * Subalgebras and subrings of this algebra (for example, the ring of n x n matrices with integral entries) arise naturally in many areas of mathemat ics. Historically however, the study of matrix algebras was preceded by the discovery of quatemions which, introduced in 1843 by Hamilton, found ap plications in the classical mechanics of the past century. Later it turned out that quaternion analysis had important applications in field theory. The al gebra of quaternions has become one of the classical mathematical objects; it is used, for instance, in algebra, geometry and topology. We will briefly focus on other examples of non-commutative rings and algebras which arise naturally in mathematics and in mathematical physics. The exterior algebra (or Grassmann algebra) is widely used in differential geometry - for example, in geometric theory of integration. Clifford algebras, which include exterior algebras as a special case, have applications in rep resentation theory and in algebraic topology. The Weyl algebra (Le. algebra of differential operators with* polynomial coefficients) often appears in the representation theory of Lie algebras. In recent years modules over the Weyl algebra and sheaves of such modules became the foundation of the so-called microlocal analysis. The theory of operator algebras (Le.

Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

This two-part EMS volume provides a succinct summary of complex algebraic geometry, coupled with a lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties. An excellent companion to the older classics on the subject.

Group theory is one of the most fundamental branches of mathematics. This highly accessible volume of the Encyclopaedia is devoted to two important subjects within this theory. Extremely useful to all mathematicians, physicists and other scientists, including graduate students who use group theory in their work.

The first contribution by Carter covers the theory of finite groups of Lie type, an important field of current mathematical research. In the second part, Platonov and Yanchevskii survey the structure of finite-dimensional division algebras, including an account of reduced K-theory.

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 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 book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.

From the reviews: "... The book under review consists of two monographs on geometric aspects of group theory ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. 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. ..." Bulletin of the London Mathematical Society, 1996

From the reviews:
..". They (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. ..." "The " "Mathematical" " Gazette, 1993"
..". The standard of this text is high and will be definitely appreciated by the algebraic community." "Monatshefte " "Mathematik, " " 1994"
.."This book is very welcome because it presents some basic material and at the same time it presents some new insights of the theory. ..." "Zentralblatt fur Mathematik, 1996""

I. Algebraic Varieties in a Projective Space.- I. Fundamental Concepts.- 1. Plane Algebraic Curves.- 1. Rational Curves.- 2. Connections with the Theory of Fields.- 3. Birational Isomorphism of Curves.- Exercises.- 2. Closed Subsets of Affine Spaces.- 1. Definition of Closed Subset.- 2. Regular Functions on a Closed Set.- 3. Regular Mappings.- Exercises.- 3. Rational Functions.- 1. Irreducible Sets.- 2. Rational Functions.- 3. Rational Mappings.- Exercises.- 4. Quasiprojective Varieties.- 1. Closed Subsets of a Projective Space.- 2. Regular Functions.- 3. Rational Functions.- 4. Examples of Regular Mappings.- Exercises.- 5. Products and Mappings of Quasiprojective Varieties.- 1. Products.- 2. Closure of the Image of a Projective Variety.- 3. Finite Mappings.- 4. Normalization Theorem.- Exercises.- 6. Dimension.- 1. Definition of Dimension.- 2. Dimension of an Intersection with a Hypersurface.- 3. A Theorem on the Dimension of Fibres.- 4. Lines on Surfaces.- 5. The Chow Coordinates of a Projective Variety.- Exercises.- II. Local Properties.- 1. Simple and Singular Points.- 1. The Local Ring of a Point.- 2. The Tangent Space.- 3. Invariance of the Tangent Space.- 4. Singular Points.- 5. The Tangent Cone.- Exercises.- 2. Expansion in Power Series.- 1. Local Parameters at a Point.- 2. Expansion in Power Series.- 3. Varieties over the Field of Real and the Field of Complex Numbers 88 Exercises.- 3. Properties of Simple Points.- 1. Subvarieties of Codimension 1.- 2. Smooth Subvarieties.- 3. Factorization in the Local Ring of a Simple Point.- Exercises.- 4. The Structure of Birational Isomorphisms.- 1. The ?-Process in a Projective Space.- 2. The Local ?-Process.- 3. Behaviour of Subvarieties under a ?-Process.- 4. Exceptional Subvarieties.- 5. Isomorphism and Birational Isomorphism.- Exercises.- 5. Normal Varieties.- 1. Normality.- 2. Normalization of Affine Varieties.- 3. Ramification.- 4. Normalization of Curves.- 5. Projective Embeddings of Smooth Varieties.- Exercises.- III. Divisors and Differential Forms.- 1. Divisors.- 1. Divisor of a Function.- 2. Locally Principal Divisors.- 3. How to Shift the Support of a Divisor Away from Points.- 4. Divisors and Rational Mappings.- 5. The Space Associated with a Divisor.- Exercises.- 2. Divisors on Curves.- 1. The Degree of a Divisor on a Curve.- 2. Bezout's Theorem on Curves.- 3. Cubic Curves.- 4. The Dimension of a Divisor.- Exercises.- 3. Algebraic Groups.- 1. Addition of Points on a Plane Cubic Curve.- 2. Algebraic Groups.- 3. Factor Groups. Chevalley's Theorem.- 4. Abelian Varieties.- 5. Picard Varieties.- Exercises.- 4. Differential Forms.- 1. One-Dimensional Regular Differential Forms.- 2. Algebraic Description of the Module of Differentials.- 3. Differential Forms of Higher Degrees.- 4. Rational Differential Forms.- Exercises.- 5. Examples and Applications of Differential Forms.- 1. Behaviour under Mappings.- 2. Invariant Differential Forms on a Group.- 3. The Canonical Class.- 4. Hypersurfaces.- 5. Hyperelliptic Curves.- 6. The Riemann-Roch Theorem for Curves.- 7. Projective Immersions of Surfaces.- Exercises.- IV. Intersection Indices.- 1. Definition and Basic Properties.- 1. Definition of an Intersection Index.- 2. Additivity of the Intersection Index.- 3. Invariance under Equivalence.- 4. End of the Proof of Invariance.- 5. General Definition of the Intersection Index.- Exercises.- 2. Applications and Generalizations of Intersection Indices.- 1. Bezout's Theorem in a Projective Space and Products of Projective Spaces.- 2. Varieties over the Field of Real Numbers.- 3. The Genus of a Smooth Curve on a Surface.- 4. The Ring of Classes of Cycles.- Exercises.- 3. Birational Isomorphisms of Surfaces.- 1. ?-Processes of Surfaces.- 2. Some Intersection Indices.- 3. Elimination of Points of Indeterminacy.- 4. Decomposition into ?-Processes.- 5. Notes and Examples.- Exercises.- II. Schemes and Varieties.- V. Schemes.- 1. Spectra of

From the reviews: ..". The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes." "Monatshefte fuer Mathematik, 1994"
..". Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory." "Medelingen van het" "wiskundig genootschap, 1995"