BRIEF REVIEWS

Edited by Ivan Netuka and Vladimír Souček. Books submitted for review should be sent to the following address: Ivan Netuka, MÚUK, Sokolovská 83, 186 00 Praha 8, Czech Republic.

C. Allday, V. Puppe: Cohomological Methods in Transformation Groups, Cambridge Studies in Advanced Mathematics 32, Cambridge University Press, Cambridge, 1993, vii+470 pp˙, GBP 50.00, ISBN 0-521-35022-0

This book deals with transformation groups. The groups in question are compact Lie groups. The authors investigate first of all the actions of tori and p-tori (p is a prime), but many results concern general compact Lie groups. As for the technique, their main tools are ordinary cohomology theory and rational homotopy theory. The aim of the authors was twofold. The book was designed firstly as a textbook providing an introduction to the subject, secondly as a convenient reference book. In both these directions, the book is very good. The reader is very carefully instructed about possible ways of reading the book, the presentation is very nice and clear, each chapter is followed by interesting exercises, and for the convenience of the reader there are attached two apendices (A: Commutative algebra, B: Some homotopy theory of differential modules). On the other hand, a substantial amount of the material contained in the book, including quite recent results, ensure that it will be a useful reference book. It is quite obvious that the authors prepared the book with great care. It makes very interesting reading and can be strongly recommended. (jiva)

Yu.G.Reshetnyak (Ed.): Geometry IV. Nonregular Riemannian Geometry, Encyclopaedia of Mathematical Sciences, vol. 70, Springer-Verlag, Berlin, 1993, 243 pp., 58 fig., DM 144.00, ISBN 3-540-54701-0, ISBN 0-387-54701-0

This volume presents a survey of results concerning generalizations of Riemannian geometries. The theory originates in the work of A. D. Aleksandrov on intrinsic geometry of convex surfaces and it is based on the later development worked out mainly by Soviet authors in the period 1945–1970. The book splits into two parts. The first part, written by Yu. G. Reshetnyak, carries the title Two-dimensional Manifolds of Bounded Curvature. The two-dimensional Riemannian manifolds are generalized in the setting of intrinsic metric spaces, i.e., spaces where any two points can be joined by a curve of a finite length and the distance of any two points is given as the maximal lower bound of the length of curves joining them. An analogue of curvature is defined for such spaces and the spaces with bounded curvature are studied. Essential parts of the theory of two-dimensional Riemannian manifolds are extended into this setting and remarkable completness results are achieved, for example the passage to a limit of a sequence of such spaces is a space of the same type under certain natural assumptions. Moreover, there are also new techniques available, first of all the results on cutting and pasting, estimates and solutions of extremal problems. In particular, an approximation of 2dimensional Riemannian manifolds by polyhedra and the translation of certain geometric problems on manifolds into problems on polyhedra is possible. The second part, Multidimensional Generalized Riemannian Spaces, is written by V. N. Berestovskij and I. G. Nikolaev. Also based on original ideas by D. A. Aleksandrov, the theory is developed here for higher dimensional spaces. The curvature is defined in terms of lengths of curves in intrinsic metric spaces and the geometry of such spaces is discussed. The exposition focuses on spaces with the curvature bounded between two constants. In particular, such spaces are Riemannian and the present approach yields an axiomatic definition of Riemannian spaces in the spirit of synthetic geometry. (jslo)

I.Kolář, P.W.Michor, J.Slovák: Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993, vi+434 pp., DM 128.00, ISBN 3-540-56235-4, ISBN 0-387-56235-4

The modern development of differential geometry established fiber bundles over manifolds as basic objects of study in differential geometry. This point of view was introduced by Nijenhuis and the present book is a monography bringing this point of view to perfection. All classical geometric objects fit in, each one can be interepreted as a natural bundle, i.e., as a map F from the category of manifolds to the category of fibred manifolds. The book presents for the first time in a book form all the available information on natural bundles and the corresponding natural operators between them. Similar questions are also studied under the name of invariant operators, the approach of the book offers a unified concept and a method for the classification of natural operators. On the way, jets spaces are used systematically as a very efficient tool. In order to be self-contained, the book includes an introduction to the basic facts of differential geometry regarded from the point of view of natural operators and a description of the Frölicher-Nijenhuis bracket, which is used then for a description of curvature and Bianchi identities. The book is very nicely and carefully written and needs almost no prerequisities. Warmly recommended to anybody interested in differential geometry. (vs)

G.Karpilovsky: Group Representations. Volume 3, North-Holland Mathematics Studies Volume 180, North-Holland, Amsterdam, 1994, xv+907 pp., $ 214.25, ISBN 0-444-87433-X

This book covers projective character theory, Clifford theory and, in part, projective representation theory. The author treats in detail Clifford theory for projective characters and pays special attention to spin representations and corresponding tables of characters. Further on, the following topics in projective representation theory are investigated: projective representations of finite groups, splitting fields for twisted group algebras, projective Schur index and projective representations of abelian groups. The second main theme is Clifford theory for graded algebras and its applications to group algebras. Here, the role of Brauer groups and G–normal algebras is emphasized. Finally, the closing part contains the study of the Schur index for ordinary representations. (tk)

V.I.Nazaikinskii, V.E.Shatalov, B.Yu.Sternin: Contact Geometry and Linear Differential Equations, Walter de Gruyter, Berlin, 1992, viii+216 pp., DM 138.00, ISBN 3-110-13381-4

It is well known that the classical method of characteristics for the solution of differential equations is very closely related to geometry. The last two decades have seen a considerable progress in this direction. The aim of this book is to make the reader familiar with important aspects of this development. The book is divided into three chapters. The first chapter has a preparatory character and brings together necessary facts both from geometry and analysis (e.gṡymplectic and contact geometry, distributions, Fourier transform). The second chapter deals with analysis. We can find there the Maslov canonical operator, Fourier-Maslov integral operators, and applications of the Maslov canonical operator theory (propagation of discontinuity, metamorphosis of discontinuity, investigation of lacunas of Green function of the Cauchy problem). The last chapter presents applications of the previous theory to the equations of principal and subprincipal types. We find there also microlocal classification of pseudodifferential operators. Reading this book needs some prerequisities from analysis and geometry but their extent is rather limited. It cannot be recommended to a complete beginner in the field because he would hardly feel sufficient motivation for these advanced investigations. The presentation also contains many quite difficult and long computations. To write this book was evidently not easy, but the result is very good. The presentation is clear, systematic and thorough. The book can be strongly recommended. (jiva)

G.Lusztig: Introduction to Quantum Groups, Progress in Mathematics, vol.110, Birkhaüser, Boston, 1993, xii+340 pp., DM 112.00, ISBN 3-764-33721-5, ISBN 0-817-63721-5

Since Drinfeld’s address to the ICM in 1986, quantum groups have been intensively studied due to their connections with various areas of mathematics and theoretical physics. There are several recently published books describing various approaches to the field and its possible applications. A special feature of the book under review is that it presents an algebraic point of view treating quantum groups as a far reaching generalizations of the classical theory of semisimple Lie algebras. Canonical bases with remarkable properties for the quantized enveloping algebras and for their modules are constructed here following methods developed by the author and by Kashiwara. Quantum analogues of Kac-Moody algebras are included as a special case. The last two parts of the book contain a discussion of special cases of roots of unity (including the case of fields of positive characteristic) and of the role of the braid group and its action in the theory. The book needs no preliminary knowledge of semisimple or Kac-Moody algebras but in one part of it, the theory of perverse sheaves (shortly reviewed in the book) is used for a construction of canonical bases. This is an important contribution to the field and can be recommended especially to mathematicians working in the field. (vs)

B.R.Gelbaum: Modern Real and Complex Analysis, J. Wiley & Sons, Inc., New York, 1995, xi+489 pp., GBP 49.95, ISBN 0-471-10715-8

This book covers the material from classical and modern mathematical analysis usually presented at the graduate and advanced undergraduate levels. Basic topology and functional analysis, integration and measure theory, holomorphic, meromorphic, entire, harmonic functions and conformal mapping. However, there are unique features which make the book different from standard text books and attractive for both students and teachers. Here are some examples: a thorough discussion of topology via open sets, via nets and via filters, uniform spaces, Brower’s Fixed Point Theorem; Daniell’s approach to integration (including e.gḞubini-Tonelli Theorems, Haar measure) and confrontation with a measure theory approach to integration; elementary exterior calculus with applications to complex function theory, singular homology of the plane via the formulae and theorems of Cauchy; Perron’s type of solution of the Dirichlet problem, Poisson’s kernels and approximate identities in Banach algebras; Bergman’s kernel functions and conformal mapping; analytic continuation, Riemann surfaces, covering spaces, sheaves, lifts, the general uniformization theorem; Thorin’s theorem, M. Riesz’s convexity theorem with applications; an invitation to the theory of several complex variables. There is also a lot of interesting exercises as well as suggestions for further reading. The book is recommended to everybody interested in mathematical analysis. (in)

W.W.D.Hodge, D.Pedoe: Methods of Algebraic Geometry. Volume I, Cambridge University Press, Cambridge, 1994, viii+440 pp., GBP 14.95, ISBN 0-521-46900-7

This is the first volume of the famous classical monograph on algebraic geometry. It was first published 1947, reprinted 1953 and 1968. The first paperback edition appeared 1968. This book is the first volume of the forthcoming second paperback edition. (jiva)

C.Tricot: Curves and Fractal Dimension, Springer-Verlag, New York, 1995, xiv+323 pp., 163 fig., DM 74.00, ISBN 0-387-94095-2, ISBN 3-540-94095-2

Those who are interested in fractals will find the book a nice introduction to the subject. Here the emphasis lies on curves and their size. The key notion is that of the fractal dimension and considerable space is given to its calculation. The book is divided into three parts. The first part deals with linear sets, their measure and dimension. Part II deals with rectifiable curves, their local geometry and with various approaches to their length. The last part is devoted to the study of nonrectifiable curves. In particular, fractal curves, curves constructed by similarities and graphs of nondifferentiable functions are discussed in detail. The book requires two years of calculus as a prerequisite. It is not written in a traditional Definition-Theorem-Proof style, thus making it readily accessible also to physicists and engineers. Many illustrations clarify the concepts studied. Bibliographical notes put the exposition into a more general context and provide the orientation for further study. The book contains both old and new mathematicss in an attractive context. It can be recommended to students and researchers in exact as well as experimental sciences. (in)

K.Nomizu, T.Sasaki: Affine Differential Geometry. Geometry of Affine Immersions, Cambridge Tracts in Mathematics, vol.111, Cambridge University Press, Cambridge, 1994, xiv+263 pp., GBP 35.00, ISBN 0-521-44177-3

The authors present a self-contained and systematic introduction to the subject from a modern point of view. They cover most of the classical concepts and results, and they also include recent developments. Additionally, relationships to other areas of differential geometry are indicated. The book starts with an introductory chapter on affine geometry and affine connections, then the classical theory of affine immersions is developed from a new point of view, including many generalizations. The third chapter focuses on important models for which many classical results are proved. The last chapter is devoted to the general geometry of affine immersions developed in the last decade. There are eleven further brief sections dealing with various concepts and results related to the main text and four appendices. An extensive bibliography including recent publications is also included. The book collects together much classical and new material in a very readable form. It should be appreciated both by experts in the area and by newcomers. (jslo)

S.A.Albeverio, R.J.Høegh/Krohn, J.A.Marion, D.H.Testard, B.S.Torrésani: Noncommutative Distributions. Unitary Representation of Gauge Groups and Algebras, Pure and Applied Mathematics, vol.175, Marcel Dekker, Inc., New York, 1993, iii+190 pp., $ 99.75, ISBN 0-824-79131-2 Let us first explain the rather unusual title. Let X be a manifold and let G = R (real numbers). Then the group G^{X} of smooth functions on X with pointwise multiplication is a (commutative) infinitedimensional group. For any distribution T on X, there is U(1)-valued map e^{i<T,.>} from the space of test functions on X. This map is a character of the group G^{X}, hence it corresponds to an irreducible unitary representation of G^{X}. Now it is easy to see that a noncommutative generalization of distributions consists of unitary irreducible representations of an infinite-dimensional group G^{X} of all smooth maps from X to a (non-commutative) group G with pointwise multiplication. Such representations are studied in the book. In more familiar terms, the group G^{X} is usually called the gauge group. As in the classical case, infinitesimal methods of Lie algebras are also used in the book even if the relation between the Lie groups and algebras is not so straightforward as in a finite-dimensional case. As a tool for constructions of examples of unitary representations of gauge groups, jet bundles are used. Two chapters are devoted to the study of energy representations. The case dim X=1 includes the well known Kac-Moody algebras and their representations. In the case of higher dimensions only a few results are available. The book is of interest both for mathematicians and physicists working in the field. (vs) P.Mattila: Geometry of Sets and Measures in Euclidean Spaces, Cambridge studies in advanced mathematics, vol.44, Cambridge University Press, Cambridge, 1995, xii+343 pp., ISBN 0-521-46576-1 This book deals mainly with the study of geometric structure of general Borel sets and Borel measures in Euclidean spaces. Chapters 1-7 develop the measure theory needed for further exposition (covering theorems, differentiation of measures, Hardy-Littlewod maximal function, invariant measures, Hausdorff measures, density theorems for Hausdorff measure as well as other measures, Rademacher and Sard-type theorems for Lipschitz maps). Chapter 8 shows how potentialtheoretic methods can be used for the study of Hausdorff dimension. Relations between capacities and Hausdorff measures are discussed. The very recent Howroyd technique is used for the proof of Frostman’s lemma as well as the proof of the existence of subsets with positive and finite Hausdorff measure. Hausdorff dimension of orthogonal projections is studied in chapter 9, plane sections in chapter 10. The local structure of sdimensional sets and measures is investigated in chapter 11. It is shown in Chapter 12 that the Fourier transform represent an efficient tool for the study of Hausdorff dimension, capacities and energy-type integrals. Chapter 13 deals with Hausdorff dimension and capacities of intersections. In Chapter 14, tangent measures in the sense of Preiss are introduced and applied to prove Marstrand’s theorem. Chapters 15 and 16 are devoted to rectifiable sets, Chapter 17 discusses the Preiss theorem characterizing rectifiability in terms of densities. Again, tangent measures turn out to be a basic tool. A characterization of rectifiability in terms of projection properties (Besicovitch, Federer) is proved in Chapter 18. The relation of rectifiability to analytic capacity and to singular integrals are studied in Chapters 19 and 20. The book is strongly recommended to analysts interested in geometric measure theory and its application. The choice of material, the presentation and the emphasis on the recent development of the subject make the book very attractive. (in)

G.Fischer: Ebene Algebraische Kurven, Vieweg Studium. Aufbaukurs Mathematik, Bd. 67, Friedrich Vieweg & Sohn, Braunschweig, 1994, x+177 pp., 156 fig., DM 38.00, ISBN 3-528-07267-9

This book covers the material for an elementary introductory course in algebraic geometry and complex analysis, with the minimum of auxiliary apparatus. The text consists of two parts. Chapters 0 - 5 contain a brief theory of algebraic curves in the complex projective plane. The explanations which are as elementary as possible are based on simple methods, especially determinants (resultants, Hessians). This part includes a motivation for projective and complex approach, tangents, singularities, points of inflection, intersections (Bézouts Theorem), dual curves, local parametrization, Plücker formulae. In the second part (Chapters 6 - 9, Appendix) more technical tools are introduced (formal power series, constructive methods of Newton and Puiseux, fractional power series, places of curves). Relations between global and local numerical invariants of curves are deduced. Many examples and figures are presented. The book can be recommended as a useful textbook for beginners, which is rather easier than “Algebraic Curves” by R J Walker. (ava)

R.Friedman, J.W.Morgan: Smooth Four-Manifolds and Complex Surfaces, Ergebnisse der Mahematik und ihrer Grenzgebiete,3.Folge,Band 27.A Series of Modern Surveys in Math., Springer-Verlag, Berlin, 1994, x+520 pp., 17 fig., DM 168.00, ISBN 3-540-57058-6, ISBN 0-387-57058-6

A classification of differentiable manifolds is still an open problem in dimension 4. A major step was taken fifteen years ago by S. Donaldson, who introduced new polynomial invariants for smooth 4-manifolds. The methods of gauge field theory used by him are very suitable for applications to algebraic surfaces. The book presents a discussion of the Donaldson polynomials of such surfaces using methods coming from algebraic geometry. The book begins with a review of the classification of surfaces and topology of elliptic surfaces. The Donaldson polynomials and their generalizations are explained in the third part. Donaldson’s theorem (quoted here without proof) is described in the fourth chapter and establishes a connection between moduli spaces of antiselfdual connections and moduli spaces of stable holomorphic bundles on an algebraic surface.

Explicit methods for computations of the Donaldson invariants are discussed in the last parts of the book together with some results concerning minimal models for smooth 4-manifolds. To read the book, different prerequisities are needed for the different parts. The first two chapters do not need much, basic facts from gauge field theory are necessary for a description of the Donaldson polynomials, basics of several complex variables and Hermitean differential geometry are used in a discussion of holomorphic vector bundles, and a knowledge of a standard introductory course of algebraic geometry is supposed for computations of invariants. The book is an important contribution to the existing literature in the field and can be well recommended to the interested mathematical audience. (vs)

G.Fayolle, V.A.Malyshev, M.V.Menshikov: Topics in the Constructive Theory of Countable Markov Chains, Cambridge University Press, Cambridge, 1995, 169 pp., GBP 27.95, ISBN 0-521-46197-9

The main goal of this book is to provide methods that allow a complete classification of countable Markov chains. The classification criteria (necessary and sufficient conditions) are based on Lyapunov functions. The book brings many new theoretical results as well as details how the Lyapunov functions are constructed. In Chapter 1 exact definitions of recurrence, transience and ergodicity are given and some classical examples are demonstrated. General classification criteria involving semimartingales are summarized in Chapter 2. Chapter 3 provides techniques for an explicit geometrical construction of Lyapunov functions. Random walks in the space Z_{+}^{2} = {(i, j) : i, j ≥ 0, integers}, and in Z_{+}^{N}, N ≥ 2, especially Markovian Jackson networks are classified by means of such constructions. Second vector fields and paths are considered in Chapter 4. Chapter 5 deals with the random scattering, Chapter 6 with the continuity of stationary distributions for families of homogeneous irreducible and aperiodic Markov chains. The exponential rate of convergence to the stationary distribution and the exponential boundedness of the limiting probabilities are established in Chapter 7. (zp)

I.J.Bakelman: Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer-Verlag, Berlin, 1994, xxi+510 pp., DM 168.00, ISBN 3-540-13620-7, ISBN 0-387-13620-7

This book is devoted to convex functions and sets, global geometric problems for partial differential equations; special attention is paid to the Monge – Ampère equations. Investigations of the boundary value problem for weak and generalized solutions of the Monge – Ampère equations are given in the second part (Chapters 3 – 6) of the book. The solutions of such problems are convex and concave functions. Questions of C^{m} – smoothness for m ≥ 2 of the weak and generalized solutions are studied. A priori bounds and geometric maximum principles for generalized and smooth solutions of the Dirichlet problem for quasilinear elliptic equations are presented together with applications (calculus of variations, differential geometry and applied mathematics) in the third part (Chapters 7 and 8) of the book. The introductory first part of the book contains a survey of elements of convex analysis. This survey is self-contained and represents a nice account of the fundamental knowledge concerning convex bodies and hypersurfaces in the first chapter and on Minkowski’s mixed volumes and some global problems concerning geometric partial differential equations in the second chapter. The book is a nice graduate text as well as a good reference source for convex analysis and its applications in the theory of partial differential equations. (ss)

P.N.Hoffman, J.F.Humphreys: Projective Representations of the Symmetric Groups. Q-Functions and Shifted Tableaux, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1992, ix+304 pp., GBP 40.00, ISBN 0-198-53556-2

The main theme of the book is a systematic discussion of the projective representations of finite groups. Attention is concentrated mostly on the symmetric and alternating groups. After five preparatory chapters, the principal results on the classification of representations of these groups (going back to Schur) are explained in the next three chapters. The last part of the book decribes the algebraic theory of Q-functions including that part of combinatorics connected with them and an explicit computation of characters. Almost half the chapters have appendices which are not used in the main text. Four of them describe an alternative way of proving Schur’s basic results. The other two describe relations among Q-functions, Schur and Hall-Littlewood functions and some recent results in modular theory. Useful comments at the end of each chapter describe the evolution of the topic and contain further references. A very nice and systematic book recommended to everybody interested in the topic. (vs)

The Uncertainty Principle in Harmonic Analysis, Ergebnisse der Math.und ihrer Grenzgebiete 3.Folge. A Series of Modern Surveys in Math., vol.28, Springer-Verlag, Berlin, 1994, vii+543 pp., 45 fig., DM 188.00, ISBN 3-540-56991-X, ISBN 0-387-56991-X

The book consists of two parts. The first and shorter part is entitled ‘The Uncertainity Principle Without Complex Variables’ (approx. 100 pages) and the second part ‘Complex Methods’ (approx. 400 pages). The authors present the book as a collection of variations on the theme: It is impossible for a non–zero function and its Fourier transform to be simultaneously very small. The first part contains a thorough discussion of the F. and M. Riesz theorem (with various proofs and generalizations), the Ivashev – Musatov theorem giving the necessary condition being zero on a charge which is singular with a finite l^{2} norm of its Fourier transform and some functional analytic approach to the Uncertainity Principle problem. The second part starts with a complex view of the Uncertainity Principle. The considerations are based on the fact that a function of a real variable with bounded spectrum coincides on R with an entire function of a complex variable. If the spectrum of a function is semibounded then this function can be analytically extended to the upper (or lower) complex halfplane. The introductory chapter of the second part of the book concludes with some information on the Cartwright class of entire functions which includes the Fourier image of any charge with compact support. The further considerations are split into two cases when the logarithmic integral converges or diverges. Missing frequencies and the diameter of the support are studied in the next chapter together with the Beurling – Malliavin theorem and the Fabry theorem. The final chapter of the second part of the book is devoted to the study of local and non–local convolution operators. The present book is a very good advanced textbook in Fourier analysis and can be helpful also for experts. (ss)

C.C.Yang, G.C.Wen, K.Y.Li, Y.M.Chiang(Eds.): Complex Analysis and its Applications, Pitman Research Notes in Mathematics Series, vol.305, Longman Scientific & Technical, Harlow, 1994, 359 pp., GBP 39.00, ISBN 0-582-23181-7

This book contains contributions presented at the international conference on complex analysis and its applications held in Hong Kong, January, 1993. Four plenary addresses (H.Begehr, C. H. Fitzgerald, R. P. Gilbert and Zhenyuan Xu, Yum-Tong Siu) and 34 contributions in the following areas of complex analysis: Value Distribution Theory (11); Complex Dynamical Systems and Geometric Function Theory (15); Applications of Complex Analysis (8). Eight pages of open problems. The chairman of the conference was Chung-Chun Yang. (bn)

J.Brüning, V.W.Guillemin (Eds.): Mathematics Past and Present. Fourier Integral Operators, Springer-Verlag, Berlin, 1994, 284 pp., DM 98.00, ISBN 0-540-56741-0, ISBN 0-387-56741-0

The study of integral operatorrs went through several important steps: singular operators of Calderón and Zygmund, pseudodifferential operators introduced by Kohn and Nirenberg and finally the Fourier integral operators studied first by Hörmander 25 years ago. The book consists of the reprints of four classical papers on the topics written by Hörmander, Duistermaat and Guillemin together with a short and very nicely written introduction to the topic by Guillemin. The book appears in a new series “Mathematics, Past and Present” introduced by Springer-Verlag. The idea of editing such books is excellent and hopefully there will be further volumes in the series. Highly recommended to anybody willing to learn quickly and efficiently basic facts on an important topic and to add a piece to his (her) mathematical culture. (vs)

L.Schneps (Ed.): The Grothedieck Theory of Dessins d’Enfants, London Mathematical Society Lecture Note Series 200, Cambridge University Press, Cambridge, 1994, 368 pp., GBP 24.95, ISBN 0-521-47821-9

In an unpublished manuscript ”Esquisse d’un Programme”, dating from 1984, A.Grothendieck sketched a remarkable relation between the Galois group G(\overline{Q}/Q) and certain combinatorial objects (“dessins”). The link between the two is provided by a theorem of Belyi which characterizes algebraic curves defined over \overline{Q} in purely “topological” terms. This book (proceedings of a conference held in Luminy in April 1993) collects 14 articles explaining and/or developing Grothendieck’s programme. It offers a comprehensive introduction to this fascinating subject that brings together as diverse topics as explicit Galois theory, non-congruence subgroups of SL_{2}(Z), braid groups, Teichmüller spaces and conformal field theory. (jnek)

O.Gloor, B.Amrhein, R.E.Maeder: Illustrierte Mathematik. Visualisierung von mathematischen Gegenständen, BirCom, Birkhäuser, Basel, 1994, 64 pp., DM 148.00, ISBN 3-764-35100-4

The publication (in German) consists of a booklet (64pp.) and a CD-ROM (about 360 MB) containing programs for Mathematica. The booklet describes the contents of the CD-ROM, its installation for different types of machines and systems, and gives a short introduction to Mathematica. Contents of the CD-ROM: geometry (polyhedra, minimal surfaces, surfaces of revolution, various curves), linear mappings, calculus (sequences, series, function theory, differentiation, integration, differential equations, iterated functions), complex functions (graphing, conformal maps). The programs may be very useful for teaching mathematics in the first two years at universities. Many results are prepared for animation (e.g., differentiation, integration, fields of solutions of differential equations). For those who do not have the program Mathematica and who do not want to create their own programs, a program Mathread is included allowing one to view and animate the prepared output. (mh)

G.W.Mackey: The Scope and History of Commutative and Noncommutative Harmonic Analysis, History of Mathematics, vol.5, American Mathematical Society, London Mathematical Society, Washington, D.C., 1992, xi+370 pp., GBP 36.00 ISBN 0-821-89903-1

The theory of representations of groups is only 100 years old, nevertheless it is one of the most beautiful and most important chapters in mathematics as well as in physics. The book presents a historical account of this theory together with its main applications in number theory, ergodic theory, quantum physics, probability theory and the theory of automorphic functions. The book consists of reprints of six survey articles written by G. Mackey during the last 20 years and an additional paper describing changes brought in by recent progress in Kac-Moody algebras, strong interaction models, exactly solvable nonlinear equations and two-dimensional lattice models in statistical physics. It can be felt that the book was written with a sense of beauty and enthusiasm in mind. A careful reading of the book brings a good reward to the reader. The book requires no prerequisities in mathematics or physics. The review papers are rare but very useful for general education and this one can be warmly recommended. (vs)

A.Kostrikin, I.R.Shafarevich (Eds.): Algebra V. Homological Algebra, Encyclopaedia of Mathematical Sciences, vol.38, Springer-Verlag, Berlin, 1994, 222 pp., DM 144.00, ISBN 3-540-53373-7, ISBN 0-387-53373-7

This volume of the Encyclopaedia of Mathematical Sciences, written by S.I.Gelfand and Yu.I.Manin, is devoted to homological algebra. The first three chapters (Complexes and Cohomology, The Language of Categories, Homology Groups in Algebra and in Geometry) contain the most classical aspects of the theory. The next two chapters describe derived and triangulated categories. The last three chapters (Mixed Hodge Structures, Perverse Sheaves, D-Modules) are devoted to geometrical applications of homological algebra to Hodge theory, to the theory of sheaves of topological spaces and to the theory of modules over rings of algebraic differential operators, emphasizing the categorical and homological aspects of these theories. The book is written in a very concise style without proofs and is sometimes difficult to read. In one small volume of 222 pages, it presents a survey of modern homological algebra with emphasis on geometrical applications and it can serve as a valuable reference. (pn)

A.N.Parshin, I.R.Shafarevich (Eds.): Algebraic Geometry IV. Linear Algebraic Groups. Invariant Theory, Encyclopaedia of Mathematical Sciences, vol.55, Springer-Verlag, Berlin, 1994, 284 pp., DM 144.00, ISBN 3-540-54682-0, ISBN 0-387-54682-0

This volume contains two expository articles: “Linear Algebraic Groups” by T.A.Springer and “Invariant Theory” by V.L.Popov and E.B.Vinberg. The first article has three chapters which give a survey of the theory of linear algebraic groups (including the structure theory of reductive groups) over, respectively, algebraically closed fields, arbitrary fields and special types of fields (finite, local, global). The second article presents the main concepts and results of geometric invariant theory: algebraic groups actions, structure of orbits, invariants of reductive groups, existence of quotients, Hilbert-Mumford theory, existence of slices, invariants of classical groups. The clear and concise style of both surveys (written by leading experts in the respective fields) and a detailed bibliography will make the book a valuable reference guide for students and practitioners of algebraic geometry alike. (jnek)

F.Borceux: Handbook of Categorical Algebra 2. Categories and Structures, Encyclopedia of Mathematics and its Applications, vol.51, Cambridge University Press, Cambridge, 1994, xvii+443 pp., GBP 50.00, ISBN 0-521-44179-X

This is the second volume of the Handbook of Categorical Algebra. The objective of these three volumes is “to give an integrated presentation of what (in author’s opinion) a graduate student should ideally know of category theory before starting research on one precise topic in this domain”. After having presented the general theory of limits, adjoint functors and Kan extensions in the first volume, in this second volume the author presents a selection of the most famous classes of “structured categories”, with the exception of toposes which appear in volume 3. The topics covered in this volume are abelian categories, regular categories, algebraic theories, monads, accessible categories, enriched category theory, topological categories and fibred categories. The book is well organized, easy to read and it contains a substantial amount of material, including many examples, counterexamples and exercises. (pn)

M.Auslander, I.Reiten, S.O.Smalø: Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995, xiv+423 pp., GBP 50.00, ISBN 0-521-41134-3

This book gives an account of the representation theory of Artin algebras, i.eȧlgebras over a commutative Artin ring R which are finitely generated R-modules. In simplest terms, the central problem of the book is to describe how a finite number of linear transformations can act simultaneously on a finite dimensional vector space. Here the language of linear algebra fails and it is necessary to use a more abstract approach involving the study of modules over finite dimensional algebras over fields or, slightly more generally, over Artin algebras. The authors concentrate on the part of the theory built around almost split sequences which presents a theoretical framework enabling a more systematic approach to the subject. The book is divided into eleven chapters. The first two chapters contain the basic material on Artin rings and algebras, chapter III provides many examples. The next four chapters (The Transpose and the Dual, Almost Split Sequences, Finite Representation Type, The Auslander-Reiten Quiver) contain basic material on almost split sequences. The last four chapters (Hereditary Algebras, Short Chains and Cycles, Stable Equivalence, Modules Determining Morphisms) are more or less independent of each other. All chapters are equipped with exercises of various degrees of complexity; at the end of the book, conjectures and open problems are listed. The book is written in a clear comprehensive style with full proofs. It can very well serve as an excellent reference as well as a textbook for graduate students. (pn)

R.F.Bass: Probabilitistic Techniques in Analysis, Probability and its Applications. A Series of the Applied Probability Trust, Springer-Verlag, New York, 1995, xii+392 pp., 12 fig., 82.00, ISBN 0-387-94387-0

As the author says, the book is concerned with the flow of ideas from probability theory to analysis. It covers those branches of analysis to which probability has contributed new results, new proofs, or new insigths. Chapter I is a minicourse in stochastic analysis: Brownian motion, martingales, stochastic calculus. Chapter II is an introduction to classical potential theory from the probability theory point of view (the Dirichlet problem, Choquet capacities, Newtonian potentials, Newtonian capacity, excessive functions, Martin boundary). The last three chapters cover results obtained mainly during the last 15-20 years and not included in monographic literature. Chapter III is devoted to the behaviour of harmonic functions in Lipschitz domains. Two proofs of the boundary Harnack principle are presented. As a consequence, the equivalence of the Martin and Euclidean boundaries is shown. Fatou theorems for Lipschitz domains are also proved and the support of harmonic measure is studied. Chapter IV deals with singular integrals (maximal functions, Hilbert and Riesz transforms, Littlewod-Paley functions, singular integral operators and Fourier multipliers, the spaces H^{1} and BMO). The last chapter is on analytic functions: L/’evy’s theorem is e.g. applied to prove Phragmén-Lindelöf theorem and the Riemann mapping theorem. Probabilistic proofs of Picard’s little theorem, the Koebe distorsion theorems, Beurling’s projection theorems, the corona theorem and other results are given. Boundary behaviour of analytic functions is studied, Makarov’s description of the Hausdorff dimension of the support of harmonic measure is presented, also the angular derivative problem is discussed. The book will be appreciated by both analysts as well as well as probabilists. It contains important results and also exercises and open problems. (in)

Khoan Vo Khac: Intégration et espaces de Lebesgue. Exercices et problèmes corrigés, Livrets d’exercices, Hermann, éditeurs des sciences et des arts, Paris, 1994, vi+131 pp., Ffrs 58.00, ISBN 2-705-66238-3

Khoan Vo Khac: Théorie de la mesure. Exercises et problèmes coorrigés, Livrets d’exercices, Hermann, éditeurs des sciences et des arts, Paris, 1993, vi+96 pp., Ffrs 58.00, ISBN 2-705-66133-5

The first booklet includes 90 exercises and problems from measure theory usually presented in standard courses. Set systems used in measure theory, elementary properties of measures, constructions of measures, the Lebesgue-Stieltjes measure and measurable mappings are covered. The second booklet includes 92 exercises and problems from integration: abstract Lebesgue integral, Lebesgue spaces, convergence of measurable functions, dual spaces to L^{p}, the Radon-Nikodym theorem, the Fubini theorem, stochastic independence.

Each of the 8 chapters of the booklet contains the basic definitions and statements of results. There are exercises on two levels: elementary and more advanced, several problems are also added. Complete solutions are provided. These books are recommendable to students as well as to university teachers. (in) F.Q.Gouveâ, N.Yui: Arithmetic of Diagonal Hypersurfaces over Finite Fields, London Mathematical Society Lecture Note Series 209, Cambridge University Press, Cambridge, 1995, xi+168 pp., GBP 19.95, ISBN 0-521-49834-1 The authors study arithmetics of diagonal hypersurfaces ν = {∑_{i=0}^{n+1} c_{i}X_{i}^{m} = 0} ⊂ P^{n+1} of dimension n over finite fields. The background to this investigation is provided by a conjecture of Lichtenbaum-Milne which relates (for n = 2d) asymptotics of the zeta function ζ(ν, s) at the central point s = d to various (partly conjectural) arithmetic invariants of ν. Using a decomposition of the motive of X into a sum of “Fermat motives”, the authors are able to compute the asymptotics quite explicitly (under certain assumptions). They also tabulate results of extensive machine computations of the zeta values. (jn) S.J.Gardiner: Harmonic Approximation, London Mathematical Society Lecture Note Series 221, Cambridge University Press, Cambridge, 1995, ix+132 pp., GBP 19.95, ISBN 0-521-49799-X The purpose of the book is to give an account of the theory of harmonic approximation in Euclidean space R^{n} (n ≥ 2) corresponding to the theory of holomorphic approximation in the complex plane connected with the names of Runge, Mergelyan, Arakelyan, Carleman and others. Harmonic approximation brings new delicate problems which attracted the attention of many researchers including the author of the book. While approximation on compact sets has a relatively long tradition, recently there has been a rapid development concerning approximation on relatively closed (not necessarily bounded) sets; the book includes many of these new results. The reader is assumed to be familiar with basic notions from potential theory; relevant facts concerning thin sets are collected in the preliminary chapter. The last chapters include applications of the subject (solvability of the Dirichlet problem on an open set in R^{n} with non-compact boundary, existence of a non-constant harmonic function R^{n} with identically vanishing Radon transform, and others). The exposition is careful, ample references and historical comments are provided, and the reader is successfully introduced to current research in the field. (jokr) A.Pasini: Diagram Geometries, Oxford University Press, Oxford, 1994, vii+488 pp., ISBN 0-198-53497-3 This book is concerned with geometries determined by an incidence graph. General notions like morphisms, covers and amalgams are developed and exploited only after many examples have been presented. The principal theme of the book is the recovery of geometries from a prescribed diagram (i.e., an information on structure of rank 2 residues). The author classifies many geometries in this spirit and finally focuses on Coxeter diagrams. The book is written very carefully and presents the current state of the art. (ad)

J.J.Rotman: An Introduction to the Theory of Groups. Fourth Edition, Graduate Texts in Mathematics, vol.148, Springer-Verlag, New York, 1995, xiv+513 pp., 37 fig., DM 108.00, ISBN 3-540-94285-8, ISBN 0-387-94285-8

Rotman’s book covers all the standard introductory topics in group theory (normal series, Sylow theorems, extensions, permutation and linear groups) and also an introduction to combinatorial group theory. The latter extends to about a quarter of book’s content and includes theorems of Kurosh and van Kampen, HNN extensions, and a chapter on word problems up to the Higman Imbedding Theorem. More advanced topics appear also in other sections (in particular in sections on cohomology, Abelian groups Mathieu groups and Steiner systems). Some of them did not appear in previous editions. The author gives geometrical explanations whenever available and points out many historical connections. There are many exercises and, in general, the exposition is relaxed and clear. (ad)

S.A.Huggett, K.P.Tod: An Introduction to Twistor Theory. Second Edition, London Mathematical Society Students Texts 4, Cambridge University Press, Cambridge, 1994, xii+178 pp., GBP 35.00, ISBN 0-521-45157-4, ISBN 0-521-45689-4

The book is the second edition of notes coming from graduate lectures on twistor theory. The style of the book is informal and easy to read. It covers the most important topics coming from twistor theory - the twistor description of the most important linear invariant equations on Minkowski space, the description of selfdual solutions of Yang-Mills field equation, the construction of non-linear gravitons and the Penrose’s definition of quasi-local momentum and angular momentum. Spinor calculus and basic facts of space-time geometry are described in the introductory chapters. A lot of useful exercises of different types (with solutions, resp. hints) are scattered throughout the book. New developments in the field and suggestions for further reading are contained in the last chapter. The book is designed both for mathematicians and physicists. It is a difficult task to write for both because they use a different language for the same things. The authors keep the balance very well and the book will be very useful as an introduction to the field for anybody interested. (vs)

C.M.Cambell, T.C.Hurley, E.F.Robertson, S.J.Tobin, J.J.Ward (Eds.): Groups ’93 Galway/St Andrews, volume 1, London Mathematical Society Lecture Note Series 211, Cambridge University Press, Cambridge, 1995, xii+303 pp., ISBN 0-521-47749-2 The Proceedings of Groups’93 Galway/St Andrews appear in two volumes. This first volume contains 23 papers written by authors with the name of the first author starting with a letter in the first half of alphabet. Papers are of different lengths and the longer papers include several valuable surveys (e.g. Reflections on virtually one-relator groups by K. Bencsath and B. Fine, Rickard equivalence and block theory by M. Broué, Computing the conjugacy classes of elements of a finite group by G. Butler, Automorphisms of nilpotent and related groups by M. R. Dixon, Lifting automorphisms by C. K. Gupta and V. Shpilrain, and Cohomological finiteness conditions by P. H. Kropholler). (ad)

K.Hulek, C.Kahn, S.H.Weintraub: Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, de Gruyter Expositions in Mathematics, vol.12, Walter de Gruyter, Berlin, 1993, xii+347 pp., 24 fig., DM 168.00, ISBN 3-110-13851-4

The subject of the book is a study of compactifications of moduli spaces of abelian surfaces with polarizations of type (1, p). In the first part, the authors construct a toroidal compactification of the moduli space of surfaces with a certain level structure and describe the character of singularities of its boundary components. A discussion of degenerations of abelian surfaces in the second part is based on the Mumford construction of degenerating abelian varieties over complete rings. The Horrocks-Mumford bundle on the four-dimensional projective space is a quite important example of rank 2 indecomposable bundle, the zero sets of its sections being called Horrocks-Mumford surfaces. In the last part of the book, the authors compare degenerations of abelian surfaces with degenerations of Horrocks-Mumford surfaces. The book is well written, nicely organized and will be useful for mathematicians interested in the subject. (vs)

M.Lawson, S.Hubbard, P.Pugh: Maths and Statistics for Business, Essential Math for Students, Longman Scientific & Technical, Harlow, 1995, x+293 pp., GBP 12.99, ISBN 0-582-23817-6

The purpose of this book is to present simple statistical and mathematical methods which can be used in business. The key areas of data presentation and statistical analysis are described in chapters 1 to 11. Here are some of them: descriptive statistics, fundamentals of probability theory, selected types of discrete and continuous probability distributions, sampling and sampling distributions, confidence intervals and hypothesis tests for a population mean, correlation and simple linear regression, time series analysis and forecasting. Chapter 12 is devoted to some types of index numbers. An introduction to linear programming is presented in chapter 13. Chapter 14 contains basic financial mathematics as simple and compound interest, effective rate of interest, present value, annuities and internal rate of return. Throughout the book, the emphasis is placed on real applications. Worked examples and many exercises are included in each chapter. The theoretical and abstract aspects of the statistical and mathematical methods are omitted. This book can be used in foundation and first year courses for the students of business, finance, accounting, economics and management. It is also suitable for selfstudy because of its informal and readable style. (jz)

P.J.Davis, R.Hersh, E.A.Marchisotto: The Mathematical Experience. Study Edition, Birkhäuser, Boston, 1995, xxi+487 pp., DM 78.00, ISBN 0-817-63739-7, ISBN 3-764-33739-7

This is the second edition of the book originally published in 1981, also by Birkhäuser. We can read in the introduction written for the first edition by Gian-Carlo Rota: “Making mathematics accessible to the educated layman, while keeping high scientific standards, has always been considered a treacherous navigation between Scylla of professional contempt and the Charybdis of public misunderstanding. Davis and Hersh have sailed across the Strait under full sail. They have opened a discussion of the mathematical experience that is inevitable for survival.” The book contains eight sections with titles such as: Selected Topics in Mathematics, Teaching and Learning, Mathematical Reality. These sections contain short articles on different subjects. Each section is accompanied by “Assignments and Problem Sets” in which there is a list of Topics to Explore, Essay Assignments, Problems and Suggested Readings and which contain a lot of material. Some subjects were previously treated by one of the authors (sometimes with co-author(s)) in Scientific American or a publication of the sort. While for a mathematician it is interesting to compare authors opinion about mathematical objects, discoveries, fields e.t.c., for a layman it is a unique chance to visit the world of mathematics. The book can be used in seminars for non-math students of Universities or teachers of mathematics at secondary schools. (jive)

R.P.Gilbert, K.Hackl (Eds.): Asymptotic Theories for Plates and Shells, Pitman Research Notes in Mathematics Series, vol.319, Longman Scientific & Technical, Harlow, 1995, 131 pp., GBP 23.00, ISBN 0-582-24875-2

This volume presents contributions to a minisymposium organized within the SIAM 40th Anniversary Meeting at Los Angeles in 1992. This minisymposium was focused on new results in the theory of plates and shells with emphasis to various advanced materials as thermoelastic, elastoplastic, anisotropic or composite ones. Typical analytical tools include therefore various asymptotic methods as formal expansion, homogenization, two-scale convergence, etc. The following contributions were included: “Constitutive equations for a hypermembrane shell” by J.L.Buchanan, “2D models of plate-like 3D elastic bodies” by D.Caillerie, “Viscosity dependent behaviour of viscoelastic porous media” by H.I.Ene and B.Vernescu, “The elastic-plastic plate” and “The two-dimensional, linear thermoelastic plate” by R.P.Gilbert and K.Hackl, “The weak solution for the generalized hyperanalytic equations of thin elastic shell theory” by R.P.Gilbert and G.-C. Wen, “On the buckling of adhesively connected von Kármán plates allowing for delamination – an eigenvalue hemivariational inequality approach” by D.Motreanu and P.D.Panagiotopoulos, and finally “Calculation of microstrains and microstresses in a thick non symmetric heterogeneous plate by homogenization” and “Calculation of microstresses in a thick heterogeneous plate by homogenization” by R.Tapiéro. The book is intended for both applied mathematicians and theoretically oriented engineers. (tr)