I.R. Shafarevich - Böcker

Group theory is one of the most fundamental branches ofmathematics. This volume of the Encyclopaedia is devoted totwo important subjects within group theory. The first partof the book is concerned with infinite groups. The authorsdeal with combinatorial group theory, free constructionsthrough group actions on trees, algorithmic problems,periodic groups and the Burnside problem, and the structuretheory for Abelian, soluble and nilpotent groups. They haveincluded the very latest developments; however, the materialis accessible to readers familiar with the basic concepts ofalgebra.The second part treats the theory of linear groups. It is agenuinely encyclopaedic survey written for non-specialists.The topics covered includethe classical groups, algebraicgroups, topological methods, conjugacy theorems, and finitelinear groups.This book will be very useful to allmathematicians,physicists and other scientists including graduate studentswho use group theory in their work.

This volume of the Encyclopaedia presents a modern approachto homological algebra, which is based on the systematic useof the terminology and ideas of derived categories andderived functors. The book contains applications ofhomological algebra to the theory of sheaves on topologicalspaces, to Hodge theory, and to the theory of sheaves ontopological spaces, to Hodge theory, and to the theory ofmodules over rings of algebraic differential operators(algebraic D-modules). The authors Gelfand and Manin explainall the main ideas of the theory of derived categories. Bothauthors are well-known researchers and the second, Manin, isfamousfor his work in algebraic geometry and mathematicalphysics. The book isan excellent reference for graduate students andresearchers in mathematics and also for physicists who usemethods from algebraic geomtry and algebraictopology.

The monograph aims at a general outline of old and new results on representations of finite-dimensional algebras. In a theory which developed rapidly during the last two decades, the lack of textbooks is the main impediment for novices. Therefore special attention is paid to the foundations, and proofs are included for statements which are elementary, serve comprehension or are scarcely available. In this manner the authors try to lead the reader up to a point where he can find his way through the original literature. The discourse is centered around the fairly complete theory of finitely-represented posets and algebras. The monograph provides many examples and all the needed background on decomposition theorems, quivers, almost split sequences and derived categories. It includes a survey on representations of tame and wild quivers, lists of critical algebras and a proof of the old conjectures of Brauer and Thrall.

The first contribution of this EMS volume on complex algebraic geometry touches upon many of the central problems in this vast and very active area of current research. While it is much too short to provide complete coverage of this subject, it provides a succinct summary of the areas it covers, while providing in-depth coverage of certain very important fields.The second part provides a brief and lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties, and partial differential equations of mathematical physics. The paper discusses the work of Mumford, Novikov, Krichever, and Shiota, and would be an excellent companion to the older classics on the subject.

This volume of the Encyclopaedia contains two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory. The first part is written by T.A. Springer, a well-known expert in the first mentioned field. He presents a comprehensive survey, which contains numerous sketched proofs and he discusses the particular features of algebraic groups over special fields (finite, local, and global). The authors of part two, E.B. Vinberg and V.L. Popov, are among the most active researchers in invariant theory. The last 20 years have been a period of vigorous development in this field due to the influence of modern methods from algebraic geometry. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

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 volume of the EMS consists of two parts. The first entitled Combinatorial Group Theory and Fundamental Groups, written by Collins and Zieschang, provides a readable and comprehensive description of that part of group theory which has its roots in topology in the theory of the fundamental group and the theory of discrete groups of transformations. Throughout the emphasis is on the rich interplay between the algebra and the topology and geometry. The second part by Grigorchuk and Kurchanov is a survey of recent work on groups relating to topological manifolds, dealing with equations in groups, particularly in surface groups and free groups, a study in terms of groups of Heegaard decompositions and algorithmic aspects of the Poincare conjecture, as well as the notion of the growth of groups. The authors have included a list of open problems, some of which have not been considered previously. Both parts contain numerous examples, outlines of proofs and full references to the literature. The book will be very useful as a reference and guide to researchers and graduate students in algebra and topology.

The first contribution covers the theory of finite groups of Lie type, which is an important field of current mathematical research. After giving the basic information Carter describes the Deligne-Lusztig method of obtaining characters of these groups using l-adic cohomology and subsequent work of Lusztig.The second part by Platonov and Yanchevskii surveys the structure of finite-dimensional division algebras and includes an account of reduced K-theory.

This book was written over a period of more than six years. Several months after we finished our work, N.1. Fel'dman (the senior author of the book) died. All additions and corrections entered after his death were made by his coauthor. The assistance of many of our colleagues was invaluable during the writing of the book. They examined parts of the manuscript and suggested many improvements, made useful comments and corrected many errors. I much have pleasure in acknowledging our great indebtedness to them. Special thanks are due to A. B. Shidlovskii, V. G. Chirskii, A.1. Galochkin and O. N. Vasilenko. I would like to express my gratitude to D. Bertrand and J. Wolfart for their help in the final stages of this book. Finally, I wish to thank Neal Koblitz for having translated this text into English. August 1997 Yu. V.Nesterenko Transcendental Numbers N.1. Fel'dman and Yu. V. Nesterenko Translated from the Russian by Neal Koblitz Contents Notation ...................................................... 9 Introduction ................................................... 11 0.1 Preliminary Remarks .................................. 11 0.2 Irrationality of J2 ..................................... 11 0.3 The Number 1C' •••••••••••••••••••••••••••••••••••••••• 13 0.4 Transcendental Numbers ............................... 14 0.5 Approximation of Algebraic Numbers .................... 15 0.6 Transcendence Questions and Other Branches of Number Theory ..................................... 16 0.7 The Basic Problems Studied in Transcendental Number Theory ....................................... 17 0.8 Different Ways of Giving the Numbers ................... 19 0.9 Methods .......................... . . . . . . . . . . . . . . 20 . . . . .

The aim of this survey, written by V.A. Iskovskikh and Yu.G. Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic vareties with an ample anticanonical divisor. Such varieties naturally appear in the birational classification of varieties of negative Kodaira dimension, and they are very close to rational ones. This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fano-Iskovskikh "double projection" method and its modifications, the vector bundles method due to S. Mukai, and the method of extremal rays. The authors discuss uniruledness and rational connectedness as well as recent progress in rationality problems of Fano varieties. The appendix contains tables of some classes of Fano varieties. This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

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

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

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 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.

Group theory is one of the most fundamental branches ofmathematics. This volume of the Encyclopaedia is devoted totwo important subjects within group theory. The first partof the book is concerned with infinite groups. The authorsdeal with combinatorial group theory, free constructionsthrough group actions on trees, algorithmic problems,periodic groups and the Burnside problem, and the structuretheory for Abelian, soluble and nilpotent groups. They haveincluded the very latest developments; however, the materialis accessible to readers familiar with the basic concepts ofalgebra.The second part treats the theory of linear groups. It is agenuinely encyclopaedic survey written for non-specialists.The topics covered includethe classical groups, algebraicgroups, topological methods, conjugacy theorems, and finitelinear groups.This book will be very useful to allmathematicians,physicists and other scientists including graduate studentswho use group theory in their work.

The value of Aw is determined up to mv + nv , where v and v are complex numbers, and m and n are 1 2 1 2 integers. The set of linear combinations mv+ nv forms a lattice H C C, and 1 2 so to each elliptic integral Aw we can associate the torus C/ H. 2 On the other hand, equation (1) defines a curve in the affine plane C = 2 2 {(z,w)}.

The problems being solved by invariant theory are far-reaching generalizations and extensions of problems on the "reduction to canonical form" of various is almost the same thing, projective geometry. objects of linear algebra or, what Invariant theory has a ISO-year history, which has seen alternating periods of growth and stagnation, and changes in the formulation of problems, methods of solution, and fields of application. In the last two decades invariant theory has experienced a period of growth, stimulated by a previous development of the theory of algebraic groups and commutative algebra. It is now viewed as a branch of the theory of algebraic transformation groups (and under a broader interpretation can be identified with this theory). We will freely use the theory of algebraic groups, an exposition of which can be found, for example, in the first article of the present volume. We will also assume the reader is familiar with the basic concepts and simplest theorems of commutative algebra and algebraic geometry; when deeper results are needed, we will cite them in the text or provide suitable references.

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 Gröbner 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.

After giving the basic information Carter describes the Deligne-Lusztig method of obtaining characters of these groups using l-adic cohomology and subsequent work of Lusztig. The second part by Platonov and Yanchevskii surveys the structure of finite-dimensional division algebras and includes an account of reduced K-theory.

This book was written over a period of more than six years. 11 0.3 The Number 1C' •••••••••••••••••••••••••••••••••••••••• 13 0.4 Transcendental Numbers ............................... 16 0.7 The Basic Problems Studied in Transcendental Number Theory ....................................... . . . . . . . . . . . . . . 20 . . . . .

This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fano-Iskovskikh "double projection" method and its modifications, the vector bundles method due to S.

This EMS volume consists of two parts. The first part is devoted to the exposition of the cohomology theory of algebraic varieties. The second part deals with algebraic surfaces. The authors have taken pains to present the material rigorously and coherently. The book contains numerous examples and insights on various topics.This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetic algebraic geometry, complex analysis and related fields.The authors are well-known experts in the field and I.R. Shafarevich is also known for being the author of volume 11 of the Encyclopaedia.