The 12 lectures presented in Representation Theories and Algebraic Geometry focus on the very rich and powerful interplay between algebraic geometry and the representation theories of various modern mathematical structures, such as reductive groups, quantum groups, Hecke algebras, restricted Lie algebras, and their companions. This interplay has been extensively exploited during recent years, resulting in great progress in these representation theories. Conversely, a great stimulus has been given to the development of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology. The range of topics covered is wide, from equivariant Chow groups, decomposition classes and Schubert varieties, multiplicity free actions, convolution algebras, standard monomial theory, and canonical bases, to annihilators of quantum Verma modules, modular representation theory of Lie algebras and combinatorics of representation categories of Harish-Chandra modules.

1. Preliminaries, Notation, and Terminology n n 1.1. Sets and functions in lR. * Throughout the book, lR. stands for the n-dimensional arithmetic space of points x = (X},X2,'" ,xn)j Ixl is the length of n n a vector x E lR. and (x, y) is the scalar product of vectors x and y in lR. , i.e., for x = (Xl, X2, *.* , xn) and y = (y}, Y2,**., Yn), Ixl = Jx~ + x~ + ...+ x~, (x, y) = XIYl + X2Y2 + ...+ XnYn. n Given arbitrary points a and b in lR. , we denote by [a, b] the segment that joins n them, i.e. the collection of points x E lR. of the form x = >.a + I'b, where>. + I' = 1 and >. ~ 0, I' ~ O. n We denote by ei, i = 1,2, ...,n, the vector in lR. whose ith coordinate is equal to 1 and the others vanish. The vectors el, e2, ...,en form a basis for the space n lR. , which is called canonical. If P( x) is some proposition in a variable x and A is a set, then {x E A I P(x)} denotes the collection of all the elements of A for which the proposition P( x) is true.

Generalising classical concepts of probability theory, the investigation of operator (semi)-stable laws as possible limit distributions of operator-normalized sums of i.i.d. random variable on finite-dimensional vector space started in 1969. Currently, this theory is still in progress and promises interesting applications. Parallel to this, similar stability concepts for probabilities on groups were developed during recent decades. It turns out that the existence of suitable limit distributions has a strong impact on the structure of both the normalizing automorphisms and the underlying group. Indeed, investigations in limit laws led to contractable groups and - at least within the class of connected groups - to homogeneous groups, in particular to groups that are topologically isomorphic to a vector space. Moreover, it has been shown that (semi)-stable measures on groups have a vector space counterpart and vice versa. The purpose of this book is to describe the structure of limit laws and the limit behaviour of normalized i.i.d. random variables on groups and on finite-dimensional vector spaces from a common point of view. This will also shed a new light on the classical situation. Chapter 1 provides an introduction to stability problems on vector spaces. Chapter II is concerned with parallel investigations for homogeneous groups and in Chapter III the situation beyond homogeneous Lie groups is treated. Throughout, emphasis is laid on the description of features common to the group- and vector space situation. Chapter I can be understood by graduate students with some background knowledge in infinite divisibility. Readers of Chapters II and III are assumed to be familiar with basic techniques from probability theory on locally compact groups.

by spin or (spin s = 1/2) field equations is emphasized because their solutions can be used for constructing solutions of other field equations insofar as fields with any spin may be constructed from spin s = 1/2 fields. A brief account of the main ideas of the book is presented in the Introduction. The book is largely based on the authors' works 55-109, 176-189, 13-16, 7*-14*,23*, 24*] carried out in the Institute of Mathematics, Academy of Sciences of the Ukraine. References to other sources is not intended to imply completeness. As a rule, only those works used directly are cited. The authors wish to express their gratitude to Academician Yu.A. Mitropoi sky, and to Academician of Academy of Sciences of the Ukraine O.S. Parasyuk, for basic support and stimulation over the course of many years; to our cowork ers in the Department of Applied Studies, LA. Egorchenko, R.Z. Zhdanov, A.G. Nikitin, LV. Revenko, V.L Lagno, and I.M. Tsifra for assistance with the manuscript."

Simplicity theory is an extension of stability theory to a wider class of structures, containing, among others, the random graph, pseudo-finite fields, and fields with a generic automorphism. Following Kim's proof of forking symmetry' which implies a good behaviour of model-theoretic independence, this area of model theory has been a field of intense study. It has necessitated the development of some important new tools, most notably the model-theoretic treatment of hyperimaginaries (classes modulo type-definable equivalence relations). It thus provides a general notion of independence (and of rank in the supersimple case) applicable to a wide class of algebraic structures. The basic theory of forking independence is developed, and its properties in a simple structure are analyzed. No prior knowledge of stability theory is assumed; in fact many stability-theoretic results follow either from more general propositions, or are developed in side remarks. Audience: This book is intended both as an introduction to simplicity theory accessible to graduate students with some knowledge of model theory, and as a reference work for research in the field.

'Et moi, .... si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alit.' human race. It has put common sense back Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bcll o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nOD linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

In Chapter 6, we describe the concept of braid equivalence from the topological point of view. This will lead us to a new concept braid homotopy that is discussed fully in the next chapter. As just mentioned, in Chapter 7, we shall discuss the difference between braid equivalence and braid homotopy. Also in this chapter, we define a homotopy braid invariant that turns out to be the so-called Milnor number. Chapter 8 is a quick review of knot theory, including Alexander's theorem. While, Chapters 9 is devoted to Markov's theorem, which allows the application of this theory to other fields. This was one of the motivations Artin had in mind when he began studying braid theory. In Chapter 10, we discuss the primary applications of braid theory to knot theory, including the introduction of the most important invariants of knot theory, the Alexander polynomial and the Jones polynomial. In Chapter 11, motivated by Dirac's string problem, the ordinary braid group is generalized to the braid groups of various surfaces. We discuss these groups from an intuitive and diagrammatic point of view. In the last short chapter 12, we present without proof one theorem, due to Gorin and Lin [GoL] , that is a surprising application of braid theory to the theory of algebraic equations.

0.1. General remarks. For any algebraic system A, the set SubA of all subsystems of A partially ordered by inclusion forms a lattice. This is the subsystem lattice of A. (In certain cases, such as that of semigroups, in order to have the right always to say that SubA is a lattice, we have to treat the empty set as a subsystem.) The study of various inter-relationships between systems and their subsystem lattices is a rather large field of investigation developed over many years. This trend was formed first in group theory; basic relevant information up to the early seventies is contained in the book [Suz] and the surveys [K Pek St], [Sad 2], [Ar Sad], there is also a quite recent book [Schm 2]. As another inspiring source, one should point out a branch of mathematics to which the book [Baer] was devoted. One of the key objects of examination in this branch is the subspace lattice of a vector space over a skew field. A more general approach deals with modules and their submodule lattices. Examining subsystem lattices for the case of modules as well as for rings and algebras (both associative and non-associative, in particular, Lie algebras) began more than thirty years ago; there are results on this subject also for lattices, Boolean algebras and some other types of algebraic systems, both concrete and general. A lot of works including several surveys have been published here.

These two volumes constitute the Proceedings of the Conference Moshe Flato, 1999'. Their spectrum is wide but the various areas covered are, in fact, strongly interwoven by a common denominator, the unique personality and creativity of the scientist in whose honor the Conference was held, and the far-reaching vision that underlies his scientific activity. With these two volumes, the reader will be able to take stock of the present state of the art in a number of subjects at the frontier of current research in mathematics, mathematical physics, and physics. Volume I is prefaced by reminiscences of and tributes to Flato's life and work. It also includes a section on the applications of sciences to insurance and finance, an area which was of interest to Flato before it became fashionable. The bulk of both volumes is on physical mathematics, where the reader will find these ingredients in various combinations, fundamental mathematical developments based on them, and challenging interpretations of physical phenomena. Audience: These volumes will be of interest to researchers and graduate students in a variety of domains, ranging from abstract mathematics to theoretical physics and other applications. Some parts will be accessible to proficient undergraduate students, and even to persons with a minimum of scientific knowledge but enough curiosity.

Every Abelian group can be related to an associative ring with an identity element, the ring of all its endomorphisms. Recently the theory of endomor phism rings of Abelian groups has become a rapidly developing area of algebra. On the one hand, it can be considered as a part of the theory of Abelian groups; on the other hand, the theory can be considered as a branch of the theory of endomorphism rings of modules and the representation theory of rings. There are several reasons for studying endomorphism rings of Abelian groups: first, it makes it possible to acquire additional information about Abelian groups themselves, to introduce new concepts and methods, and to find new interesting classes of groups; second, it stimulates further develop ment of the theory of modules and their endomorphism rings. The theory of endomorphism rings can also be useful for studies of the structure of additive groups of rings, E-modules, and homological properties of Abelian groups. The books of Baer [52] and Kaplansky [245] have played an important role in the early development of the theory of endomorphism rings of Abelian groups and modules. Endomorphism rings of Abelian groups are much stu died in monographs of Fuchs [170], [172], and [173]. Endomorphism rings are also studied in the works of Kurosh [287], Arnold [31], and Benabdallah [63].

This volume contains the proceedings of the First Ukrainian-French Romanian School "Algebraic and Geometric Methods in Mathematical Physics," held in Kaciveli, Crimea (Ukraine) from 1 September ti1114 September 1993. The School was organized by the generous support of the Ministry of Research and Space of France (MRE), the Academy of Sciences of Ukraine (ANU), the French National Center for Scientific Research (CNRS) and the State Committee for Science and Technologies of Ukraine (GKNT). Members of the International Scientific Committee were: J.-M. Bony (paris), A. Boutet de Monvel-Berthier (Paris, co-chairman), P. Cartier (paris), V. Drinfeld (Kharkov), V. Georgescu (Paris), J.L. Lebowitz (Rutgers), V. Marchenko (Kharkov, co-chairman), V.P. Maslov (Moscow), H. Mc-Kean (New-York), Yu. Mitropolsky (Kiev), G. Nenciu (Bucharest, co-chairman), S. Novikov (Moscow), G. Papanicolau (New-York), L. Pastur (Kharkov), J.-J. Sansuc (Paris). The School consisted of plenary lectures (morning sessions) and special sessions. The plenary lectures were intended to be accessible to all participants and plenary speakers were invited by the scientific organizing committee to give reviews of their own field of interest. The special sessions were devoted to a variety of more concrete and technical questions in the respective fields. According to the program the plenary lectures included in the volume are grouped in three chapters. The fourth chapter contains short communications."

This book has developed from a series of lectures which were given by the author in mechanics-mathematics department of the Moscow State University. In 1981 the course "Additional chapters in algebra" replaced the course "Gen eral algebra" which was founded by A. G. Kurosh (1908-1971), professor and head of the department of higher algebra for a period of several decades. The material of this course formed the basis of A. G. Kurosh's well-known book "Lectures on general algebra" (Moscow,1962; 2-nd edition: Moscow, Nauka, 1973) and the book "General algebra. Lectures of 1969-1970. " (Moscow, Nauka, 1974). Another book based on the course, "Elements of general al gebra" (M.: Nauka, 1983) was published by L. A. Skorniakov, professor, now deceased, in the same department. It should be noted that A. G. Kurosh was not only the lecturer for the course "General algebra" but he was also the recognized leader of the scientific school of the same name. It is difficult to determine the limits of this school; however, the "Lectures . . . " of 1962 men tioned above contain some material which exceed these limits. Eventually this effect intensified: the lectures of the course were given by many well-known scientists, and some of them see themselves as "general algebraists." Each lecturer brought significant originality not only in presentation of the material but in the substance of the course. Therefore not all material which is now accepted as necessary for algebraic students fits within the scope of general algebra."

Lattice-valued Logic aims at establishing the logical foundation for uncertain information processing routinely performed by humans and artificial intelligence systems. In this textbook for the first time a general introduction on lattice-valued logic is given. It systematically summarizes research from the basic notions up to recent results on lattice implication algebras, lattice-valued logic systems based on lattice implication algebras, as well as the corresponding reasoning theories and methods. The book provides the suitable theoretical logical background of lattice-valued logic systems and supports newly designed intelligent uncertain-information-processing systems and a wide spectrum of intelligent learning tasks.

A description of the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I provides an introduction to the geometry of geodesic spaces, while Part II develops the basic theory of spaces with upper curvature bounds. More specialized topics, such as complexes of groups, are covered in Part III.

Recent major advances in model theory include connections between model theory and Diophantine and real analytic geometry, permutation groups, and finite algebras. The present book contains lectures on recent results in algebraic model theory, covering topics from the following areas: geometric model theory, the model theory of analytic structures, permutation groups in model theory, the spectra of countable theories, and the structure of finite algebras. Audience: Graduate students in logic and others wishing to keep abreast of current trends in model theory. The lectures contain sufficient introductory material to be able to grasp the recent results presented.

Our prime concern in this book is to discuss some most interesting prosppcts that have occurred recently in conformally invariant quantum field theory in a D-diuwnsional space. One of the most promising trends is constructing an pxact solution for a cprtain class of models. This task seems to be quite feasible in the light of recent resllits. The situation here is to some extent similar to what was going on in the past ypars with the two-dimensional quantum field theory. Our investigation of conformal Ward identities in a D-dimensional space, carried out as far hack as the late H. J7Gs, showed that in the D-dimensional quantum field theory, irrespective of the type of interartion, there exists a special set of states of the field with the following property: if we rpqllire that one of these states should vanish, this determines an exact solution of 3. certain field model. These states are analogous to null-vectors which determine the minimal models in the two-dimensional field theory. On the other hand, the recent resparches supplied us with a number of indications on the existencp of an intinite-parampter algebra analogous to the Virasoro algebra in spaces of higher dimensions D 2: :~. It has also been shown that this algebra admits an operator rentral expansion. It seems to us that the above-mentioned models are field theoretical realizations of the representations of these new symmetries for D 2: ;3.

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. Two chapters provide complete and simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

Helmut Koch's classic is now available in English. Competently translated by Franz Lemmermeyer, it introduces the theory of pro-p groups and their cohomology. The book contains a postscript on the recent development of the field written by H. Koch and F. Lemmermeyer, along with many additional recent references.

X Kochendorffer, L.A. Kalu: lnin and their students in the 50s and 60s. Nowadays the most deeply developed is the theory of binary invariant relations and their combinatorial approximations. These combinatorial approximations arose repeatedly during this century under various names (Hecke algebras, centralizer rings, association schemes, coherent configurations, cellular rings, etc.-see the first paper of the collection for details) andin various branches of mathematics, both pure and applied. One of these approximations, the theory of cellular rings (cellular algebras), was developed at the end of the 60s by B. Yu. Weisfeiler and A.A. Leman in the course of the first serious attempt to study the complexity of the graph isomorphism problem, one of the central problems in the modern theory of combinatorial algorithms. At roughly the same time G.M. Adelson-Velskir, V.L. Arlazarov, I.A. Faradtev and their colleagues had developed a rather efficient tool for the constructive enumeration of combinatorial objects based on the branch and bound method. By means of this tool a number of "sports-like" results were obtained. Some of these results are still unsurpassed."

The present volume has its origins in a pair of informal workshops held at the Free University of Brussels, in June of 1998 and May of 1999, named "Current Research 1 in Operational Quantum Logic." These brought together mathematicians and physicists working in operational quantum logic and related areas, as well as a number of interested philosophers of science, for a rare opportunity to discuss recent developments in this field. After some discussion, it was decided that, rather than producing a volume of conference proceedings, we would try to organize the conferees to produce a set of comprehensive survey papers, which would not only report on recent developments in quantum logic, but also provide a tutorial overview of the subject suitable for an interested non-specialist audience. The resulting volume provides an overview of the concepts and methods used in current research in quantum logic, viewed both as a branch of mathemati cal physics and as an area of pure mathematics. The first half of the book is concerned with the algebraic side of the subject, and in particular the theory of orthomodular lattices and posets, effect algebras, etc. In the second half of the book, special attention is given to categorical methods and to connections with theoretical computer science. At the 1999 workshop, we were fortunate to hear three excellent lectures by David J. Foulis, represented here by two contributions. Dave's work, spanning 40 years, has helped to define, and continues to reshape, the field of quantum logic."

This book, in some sense, began to be written by the first author in 1983, when optional lectures on Abelian groups were held at the Fac ulty of Mathematics and Computer Science, 'Babes-Bolyai' University in Cluj-Napoca, Romania. From 1992, these lectures were extended to a twosemester electivecourse on abelian groups for undergraduate stu dents, followed by a twosemester course on the same topic for graduate students in Algebra. All the other authors attended these two years of lectures and are now Assistants to the Chair of Algebra of this Fac ulty. The first draft of this collection, including only exercises solved by students as home works, the last ten years, had 160pages. We felt that there is a need for a book such as this one, because it would provide a nice bridge between introductory Abelian Group Theory and more advanced research problems. The book InfiniteAbelianGroups, published by LaszloFuchsin two volumes 1970 and 1973 willwithout doubt last as the most important guide for abelian group theorists. Many exercises are selected from this source but there are plenty of other bibliographical items (see the Bibliography) which were used in order to make up this collection. For some of the problems stated, recent developments are also given. Nevertheless, there are plenty of elementary results (the so called 'folklore') in Abelian Group Theory whichdo not appear in any written material. It is also one purpose of this book to complete this gap."

This volume contains one invited lecture which was presented by the 1994 Fields Medal ist Professor E. Zelmanov and twelve other papers which were presented at the Third International Conference on Algebra and Their Related Topics at Chang Jung Christian University, Tainan, Republic of China, during the period June 26-July 1, 200l. All papers in this volume have been refereed by an international referee board and we would like to express our deepest thanks to all the referees who were so helpful and punctual in submitting their reports. Thanks are also due to the Promotion and Research Center of National Science Council of Republic of China and the Chang Jung Christian University for their generous financial support of this conference. The spirit of this conference is a continuation of the last two International Tainan Moscow Algebra Workshop on Algebras and Their Related Topics which were held in the mid-90's of the last century. The purpose of this very conference was to give a clear picture of the recent development and research in the fields of different kinds of algebras both in Taiwan and in the rest ofthe world, especially say, Russia" Europe, North America and South America. Thus, we were hoping to enhance the possibility of future cooperation in research work among the algebraists ofthe five continents. Here we would like to point out that this algebra gathering will constantly be held in the future in the southern part of Taiwan."

This is a book about representing symmetry in quantum mechanics. The book is on a graduate and/or researcher level and it is written with an attempt to be concise, to respect conceptual clarity and mathematical rigor. The basic structures of quantum mechanics are used to identify the automorphism group of quantum mechanics. The main concept of a symmetry action is defined as a group homomorphism from a given group, the group of symmetries, to the automorphism group of quantum mechanics. The structure of symmetry actions is determined under the assumption that the symmetry group is a Lie group. The Galilei invariance is used to illustrate the general theory by giving a systematic presentation of a Galilei invariant elementary particle. A brief description of the Galilei invariant wave equations is also given.

These two volumes constitute the Proceedings of the Conference Moshe Flato, 1999'. Their spectrum is wide but the various areas covered are, in fact, strongly interwoven by a common denominator, the unique personality and creativity of the scientist in whose honor the Conference was held, and the far-reaching vision that underlies his scientific activity. With these two volumes, the reader will be able to take stock of the present state of the art in a number of subjects at the frontier of current research in mathematics, mathematical physics, and physics. Volume I is prefaced by reminiscences of and tributes to Flato's life and work. It also includes a section on the applications of sciences to insurance and finance, an area which was of interest to Flato before it became fashionable. The bulk of both volumes is on physical mathematics, where the reader will find these ingredients in various combinations, fundamental mathematical developments based on them, and challenging interpretations of physical phenomena. Audience: These volumes will be of interest to researchers and graduate students in a variety of domains, ranging from abstract mathematics to theoretical physics and other applications. Some parts will be accessible to proficient undergraduate students, and even to persons with a minimum of scientific knowledge but enough curiosity."

Asymptotic methods of nonlinear mechanics developed by N. M. Krylov and N. N. Bogoliubov originated new trend in perturbation theory. They pene- trated deep into various applied branches (theoretical physics, mechanics, ap- plied astronomy, dynamics of space flights, and others) and laid the founda- tion for lrumerous generalizations and for the creation of various modifications of thesem. E!f,hods. A great number of approaches and techniques exist and many differen. t classes of mathematical objects have been considered (ordinary differential equations, partial differential equations, delay diffe,'ential equations and others). The stat. e of studying related problems was described in mono- graphs and original papers of Krylov N. M. , Bogoliubov N. N. [1], [2], Bogoli- ubov N. N [1J, Bogoliubov N. N. , Mitropolsky Yu. A. [1], Bogoliubov N. N. , Mitropol- sky Yu. A. , Samoilenko A. M. [1], Akulenko L. D. [1], van den Broek B. [1], van den Broek B. , Verhulst F. [1], Chernousko F. L. , Akulenko L. D. and Sokolov B. N. [1], Eckhause W. [l], Filatov A. N. [2], Filatov A. N. , Shershkov V. V. [1], Gi- acaglia G. E. O. [1], Grassman J. [1], Grebennikov E. A. [1], Grebennikov E. A. , Mitropolsky Yu. A. [1], Grebennikov E. A. , Ryabov Yu. A. [1], Hale J . K. [I]' Ha- paev N. N. [1], Landa P. S. [1), Lomov S. A. [1], Lopatin A. K. [22]-[24], Lykova O. B.