1.1. From the Preface.

The present work is the first volume of a substantially enlarged version of the mimeographed notes of a course of lectures first given by me in the Indian Statistical Institute, Calcutta, India, during 1964-65. When it was suggested that these lectures be developed into a book, I readily agreed and took the opportunity to extend the scope of the material covered.

No background in physics is in principle necessary for understanding the essential ideas in this work. However, a high degree of mathematical maturity is certainly indispensable. It is safe to say that I aim at an audience composed of professional mathematicians, advanced graduate students, and, hopefully, the rapidly increasing group of mathematical physicists who are attracted to fundamental mathematical questions.

Over the years, the mathematics of quantum theory has become more abstract and, consequently, simpler. Hilbert spaces have been used from the very beginning and, after Weyl and Wigner, group representations have come in conclusively. Recent discoveries seem to indicate that the role of group representations is destined for further expansion, not to speak of the impact of the theory of several complex variables and function-space analysis. But all of this pertains to the world of interacting subatomic particles; the more modest view of the microscopic world presented in this book requires somewhat less. The reader with a knowledge of abstract integration, Hilbert space theory, and topological groups will find the going easy.

Part of the work which went into the writing of this book was supported by the National Science Foundation Grant No. GP-5224. I have profited greatly from conversations with many friends and colleagues at various institutions. To all of them, especially to R Arens, R J Blattner, R Ranga Rao, K R Parthasarathy, and S R. S Varadhan, my sincere thanks. I want to record my deep thanks to my colleague Don Babbitt who read through the manuscript carefully, discovered many mistakes, and was responsible for significant improvement of the manuscript. My apologies are due to all those whose work has been ignored or, possibly, incorrectly (and/or insufficiently) discussed. Finally, I want to acknowledge that this book might never have made its way into print but for my wife. She typed the entire manuscript, encouraged me when my enthusiasm went down, and made me understand some of the meaning of our ancient words,

You have only right to your actions, never to the fruits.

To her my deep gratitude.

1.2. Review by: P Winternitz.
The Mathematical Gazette 54 (388) (1970), 202-203.

The reviewed book is the first of two volumes devoted to the mathematics underlying modern quantum theory. The main aim of the book is to present a geometrical view of the quantum mechanical world, stressing that the basic principles are geometric in character. Essentially this is an exposition of an axiomatic approach to quantum mechanics, which goes back to the classic investigations of Dirac, von Neumann and Wigner and to which the author has contributed in a series of papers.

As the title "Projective Geometry" suggests, part 1 of the present volume is almost purely mathematical and in it Prof Varadarajan develops the mathematical tools to be used further. It is assumed that the reader has a good background in higher mathematics, specially in abstract integration, Hilbert space theory and topological groups. The starting point is the discussion of a classical system. A Boolean algebra of subsets of the phase space, called the logic of the system, is introduced (the elements of the algebra are statements about the physical system). This approach is then generalised to quantum mechanical systems, where logics are not Boolean algebras but certain projective geometries. The remaining four chapters of Part 1 are devoted to the theory of the corresponding vector geometries. These geometric concepts are applied in Part 2 to analyse the structure of the logic of quantum mechanical systems.

The customary concepts of quantum mechanics, like self-adjoint operators corresponding to observables and vectors in a Hilbert space corresponding to states of the system, are then introduced on the basis of the considered abstract logics, provided that these are projective logics in a sense defined. Thus the usually inductive description of the microworld is supplemented by a deductive one.

The book is a contribution to a branch of mathematical physics which is currently receiving a large amount of interest, namely the axiomatic foundations of contemporary physics. Although it only treats quantum mechanics at the atomic level, it will be mainly of interest for theoretical physicists, working in quantum field theory and elementary particle theory in general, where the problem of obtaining general results from basic principles is, in the absence of a real dynamic theory, of the utmost importance.

As written, the book is more accessible to mathematicians than physicists, since it assumes more mathematical sophistication than physical background. It will clearly be useful both for specialists working in the given field and for mathematicians wishing to contribute to our understanding of the physical world.

The second volume, as yet unpublished, treating the problems of a covariant with respect to certain symmetry groups, description of quantal systems, will be of special interest for active research workers.

1.3. Review by: Stanley P Gudder.
Mathematical Reviews MR0471674 (57 #11399).

This volume is concerned with the quantum logic approach to axiomatic quantum mechanics with emphasis on the geometric aspects of the subject. The book stresses a careful mathematical delineation of the foundations of quantum mechanics. This work has been very influential in later research and even though it is over ten years old, it is still frequently cited.

J von Neumann and G Birkhoff initiated this subject in their pioneering paper (1936). Stimulated by the important work of G Mackey [The mathematical foundations of quantum mechanics, 1963, C Piron (1964), and the author (1962), this field has attracted a large number of investigators. J Jauch's book [Foundations of quantum mechanics, 1968] has some overlap with the present one, although the former is more physical and less mathematical.

The volume is divided into two parts. In the first part (Projective geometry), a beautiful proof of the fundamental theorem of projective geometry is given. At the same time those areas of projective geometry which are useful to the subject at hand are developed. In the second part (The logic of quantum mechanics), a primitive axiomatic system for quantum mechanics is constructed. Basic quantum mechanical concepts such as propositions, observables, states, the supposition principle, and simultaneous observability are defined, discussed, and their properties derived. Finally, the ideas in the first part of this volume are applied to prove the Hilbert space representation theorem due mainly to C Piron.

2.1. Review by: Stanley P Gudder.
Mathematical Reviews MR0471675 (57 #11400).

The flavour of the second volume of this two volume work is quite different from that of the first [see 1.1 above]. Whereas the first volume stresses the foundations of the subject, the second moves toward the frontiers. The main concern is with the description of relativistic and nonrelativistic quantum systems with a finite number of degrees of freedom. The results of Volume 1 are used, and it is assumed that the quantum logic is a standard quantum logic on Hilbert space.

A central role is played by the symmetry group of the system. In the relativistic case this is the inhomogeneous Lorentz group, and in the nonrelativistic case it is the inhomogeneous Galilean group. The material again divides naturally into two parts. The first part is a mathematical treatment of locally compact groups and their representations. The work of G Mackey [(1949), (1952), (1953)] on systems of imprimitivity is especially important for this treatment. In the second part, results of the first part are applied to the study of covariant quantum systems, in particular relativistic and nonrelativistic particles. Here there is a unifying discussion of kinematics, dynamics, and the Schrödinger equation.

3.1. From the Preface.

This book has grown out of a set of lecture notes I had prepared for a course on Lie groups in 1966. When I lectured again on the subject in 1972, I revised the notes substantially. It is the revised version that is now appearing in book form.

The theory of Lie groups plays a fundamental role in many areas of mathematics. There are a number of books on the subject currently available - most notably those of Chevalley, Jacobson, and Bourbaki - which present various aspects of the theory in great depth. However, I feel there is a need for a single book in English which develops both the algebraic and analytic aspects of the theory and which goes into the representation theory of semi-simple Lie groups and Lie algebras in detail. This book is an attempt to fill this need. It is my hope that this book will introduce the aspiring graduate student as well as the nonspecialist mathematician to the fundamental themes of the subject.

I have made no attempt to discuss infinite-dimensional representations. This is a very active field, and a proper treatment of it would require another volume (if not more) of this size. However, the reader who wants to take up this theory will find that this book prepares him reasonably well for that task.

I have included a large number of exercises. Many of these provide the reader opportunities to test his understanding. In addition I have made a systematic attempt in these exercises to develop many aspects of the subject that could not be treated in the text: homogeneous spaces and their cohomologies, structure of matrix groups, representations in polynomial rings, and complexifications of real groups, to mention a few. In each case the exercises are graded in the form of a succession of (locally simple, I hope) steps, with hints for many. Substantial parts of Chapters 2, 3 and 4, together with a suitable selection from the exercises, could conceivably form the content of a one year graduate course on Lie groups. From the student's point of view the prerequisites for such a course would be a one-semester course on topological groups and one on differentiable manifolds.

The book begins with an introductory chapter on differentiable and analytic manifolds. A Lie group is at the same time a group and a manifold, and the theory of differentiable manifolds is the foundation on which the subject should be built. It was not my intention to be exhaustive, but I have made an effort to treat the main results of manifold theory that are used subsequently, especially the construction of global solutions to involutive systems of differential equations on a manifold. In taking this approach I have followed Chevalley, whose Princeton book was the first to develop the theory of Lie groups globally. My debt to Chevalley is great not only here but throughout the book, and it will be visible to anyone who, like me, learned the subject from his books.

The second chapter deals with the general theory. All the basic results and concepts are discussed: Lie groups and their Lie algebras, the correspondence between subgroups and subalgebras, the exponential map, the Campbell-Hausdorff formula, the theorems known as the fundamental theorems of Lie, and so on.

The third chapter is almost entirely on Lie algebras. The aim is to examine the structure of a Lie algebra in detail. With the exception of the last part of this chapter, where applications are made to the structure of Lie groups, the action takes place over a field of characteristic zero. The main results are the theorems of Lie and Engel on nilpotent and solvable algebras; Cartan's criterion for semisimplicity, namely that a Lie algebra is semisimple if and only if its Cartan-Killing form is nonsingular; Weyl's theorem asserting that all finite-dimensional representations of a semisimple Lie algebra are semisimple; and the theorems of Levi and Mal'cev on the semidirect decompositions of an arbitrary Lie algebra into its radical and a (semisimple) Levi factor. Although the results of Weyl and Levi-Mal'cev are cohomological in their nature (at least from the algebraic point of view), I have resisted the temptation to discuss the general cohomology theory of Lie algebras and have confined myself strictly to what is needed (ad hoc low-dimensional cohomology).

The fourth and final chapter is the heart of the book and is a fairly complete treatment of the fine structure and representation theory of semisimple Lie algebras and Lie groups. The root structure and the classification of simple Lie algebras over the field of complex numbers are obtained. As for representation theory, it is examined from both the infinitesimal (Cartan, Weyl, Harish-Chandra, Chevalley) and the global (Weyl) points of view. First I present the algebraic view, in which universal enveloping algebras. left ideals, highest weights, and infinitesimal characters are put in the foreground. I have followed here the treatment of Harish-Chandra given in his early papers and used it to prove the bijective nature of the correspondence between connected Dynkin diagrams and simple Lie algebras over the complexes. This algebraic part is then followed up with the transcendental theory. Here compact Lie groups come to the fore. The existence and conjugacy of their maximal tori are established, and Weyl's classic derivation of his great character formula is given. It is my belief that this dual treatment of representation theory is not only illuminating but even essential and that the infinitesimal and global parts of the theory are complementary facets of a very beautiful and complete picture.

In order not to interrupt the main flow of exposition, I have added an appendix at the end of this chapter where I have discussed the basic results of finite reflection groups and root systems. This appendix is essentially the same as a set of unpublished notes of Professor Robert Steinberg on the subject, and I am very grateful to him for allowing me to use his manuscript.

It only remains to thank all those without whose help this book would have been impossible. I am especially grateful to Professor I. M. Singer for his help at various critical stages. Mrs. Alice Hume typed the entire manuscript, and I cannot describe my indebtedness to the great skill, tempered with great patience, with which she carried out this task. I would like to thank Joel Zeitlin, who helped me prepare the original 1966 notes; and Mohsen Pazirandeh and Peter Trombi, who looked through the entire manuscript and corrected many errors. I would also like to thank Ms. Judy Burke, whose guidance was indispensable in preparing the manuscript for publication.

I would like to end this on a personal note. My first introduction to serious mathematics was from the papers of Harish-Chandra on semisimple Lie groups, and almost everything I know of representation theory goes back either to his papers or the discussions I have had with him over the past years. My debt to him is too immense to be detailed.

3.2. Review by: Shingo Murakami.
Mathematical Reviews MR0376938 (51 #13113).

The present book is designed to introduce the theory of Lie groups as a whole. In the preface the author states: "There are a number of books on the subject currently available - most notably those of Chevalley, Jacobson and Bourbaki - which present various aspects of the theory in great depth. However, I feel there is a need for a single book in English which develops both the algebraic and analytic aspects of the theory and which goes into the representation theory of semisimple Lie groups and Lie algebras in detail. This book is an attempt to fill this need." With this intention, the author has successfully organised the principal themes of Lie theory into a self-contained graduate-level text. The book is especially suitable for the graduate student as well as the nonspecialist mathematician having some acquaintance with topological groups and differentiable manifolds who wishes to study Lie theory from the beginning in considerable depth.

The book consists of four chapters, each of which is divided into several sections. Chapter 1 is a quick introduction to the theory of differentiable manifolds, and provides fundamental notions and results from that theory which are required in the following chapters. The author follows the method of presentation originated by C. Chevalley [Theory of Lie groups, 1946] and proves Frobenius' theorem on the integration of involutive systems of differential equations.

Chapter 2 deals with the general theory of Lie groups and discusses the following basic results and concepts: Lie groups and their Lie algebras, the correspondence between subgroups and subalgebras, the exponential map, local isomorphism classes of Lie groups, the Baker-Campbell-Hausdorff formula and so on. In the final section, Lie's theory of transformation groups is treated from both the local and global aspects, thus introducing a classical theorem of Lie as well as relatively recent results of R. S. Palais [A global formulation of the Lie theory of transformation groups, 1957].

Chapter 3 is devoted almost entirely to the structure theory of Lie algebras. After a review of linear algebra, the author develops systematically the theory of universal enveloping algebras, which serves as a basic tool in the subsequent discussions. Then appear the theorems of Lie and Engel on nilpotent and solvable Lie algebras, Cartan's criterion for semisimplicity, Weyl's theorem on the complete reducibility of linear representations of semisimple Lie algebras, the theorem of Levi and Malʹcev and the theorem of Ado. The existence of a simply connected Lie group corresponding to a given Lie algebra is then proved by applying the Levi decomposition of the algebra. The final section introduces a number of group-theoretical consequences of structure theory, and one finds a proof, attributed to G Hochschild [The structure of Lie groups, 1965], of the theorem of Malʹcev asserting that a normal subgroup of a simply connected Lie group is simply connected.

Chapter 4 provides a fairly complete treatment of the structure and representation theory of complex semisimple Lie algebras and Lie groups. First the structure theory is developed in full detail. After fixing a complex semisimple Lie algebra, the author defines its Cartan subalgebras, roots and root subspaces, Weyl group, Weyl basis, Cartan matrix and Dynkin diagram and examines their properties closely. In a later section these notions are illustrated for each of the classical Lie algebras. Now, the author introduces both the algebraic and the transcendental viewpoints of representation theory. First, Cartan's theorem parameterising the irreducible linear representations of a complex semisimple Lie algebra by the dominant integral forms on a Cartan subalgebra is proved in an algebraic way, after Chevalley and Harish-Chandra. Here the universal enveloping algebra and its left ideals play essential roles and, as an advantage of this method, the bijectivity of the correspondence between complex simple Lie algebras and connected Dynkin diagrams is established in the same way. The author also treats invariant polynomials on a complex semisimple Lie algebra and derives Harish-Chandra's formula for infinitestimal characters. This algebraic part is then followed by the classical transcendental theory of Weyl, which is developed as follows: Relations between compact and complex semisimple Lie groups via the "unitarian trick", maximal tori in compact semisimple Lie groups, an integral formula on compact groups and Weyl's character formula. This chapter ends with an appendix on finite reflection groups, where the main results on these groups and their representations in polynomial rings are presented, based on unpublished notes of R Steinberg.

Each of the chapters so far reviewed is complemented by a large number of exercises. Some of these may help the reader to master the subjects, but many others are presented, often with references, in order that the reader may become acquainted with interesting topics in Lie theory that are not included in the text. Among others, there are the following topics: Homogeneous spaces and their cohomology, the structure of matrix groups, real analytic groups admitting faithful representations, complexifications of real groups, an algebraic proof of Weyl's character formula and the orbit structure of the adjoint action of a complex semisimple Lie group.

Throughout this book, the terminology and notation used by the author are standard, and the theorems are often preceded by a set of lemmas and are therefore demonstrated in an easily comprehensible way. A number of examples are discussed in the text as well as in the exercises and they should greatly facilitate the reader's understanding; in particular, the detailed discussions of the classical Lie algebras in Chapter 4 may be quite useful for understanding fundamental notions about semisimple Lie algebras that are defined in a general abstract form and thus seem complicated to the novice. On the other hand, many materials treated in the book are carefully arranged. In referring to results of individual mathematicians, the author generally respects the presentation found in the original articles, and this is particularly the case in introducing early works of Harish-Chandra. With the significant extension of the text by many exercises, this book should therefore provide a good preparation and a strong motivation for further study of Lie theory, especially for the study of its most striking area - the theory of unitary representations of Lie groups.

4.1. From the Preface.

The contents of these notes are essentially the same as those of a Seminar on semisimple groups that I conducted during 1969-1973 at the University of California at Los Angeles. I am very grateful to Professors Gangolli and Eckmann for suggesting that this material appear in the Springer Lecture Notes Series and encouraging me to prepare them for publication.

My aim here has been to give a more or less self-contained exposition of Harish-Chandra's work on harmonic analysis on real reductive groups, leading to the complete determination of the discrete series. I have kept quite close to his view of the subject although the informed reader may perceive departures in detail here and there.

These notes are in two parts. Part one deals with the problems of invariant analysis on a real reductive Lie algebra. It contains a full treatment of regular orbital integrals and their Fourier transforms; it presents a detailed proof of the theorem that invariant eigendistributions are locally integrable functions; and concludes with the proof of the theorem that an analytic invariant differential operator that kills all invariant $C^{∞}$ functions, kills all invariant distributions. Part two treats the theory on the group, with descent to Lie algebra playing a key role in many proofs. Here I have proved that invariant eigendistributions on real reductive groups are locally integrable functions, given the explicit construction of the characters of the discrete series, and treated all the aspects of Schwartz space and tempered distributions that are needed to reach the goals I set out with.

Due to obvious limitations I have not made any attempt to discuss other contributions to this subject, such as orbital integrals of nilpotents, invariant analysis over local fields, to mention a few. The subject is in a very active phase of development and many recent contributions suggest a real possibility of a significantly different way of treating some of these questions. However, I feel that an exposition that attempts to maintain the original and pioneering perspective of Harish-Chandra deserves a place in the literature.

I wish to thank all my friends with whom I have discussed this subject over the past several years. In addition, I would like to thank Peter Trombi, King Lai, Mohsen Pazirandeh and Thomas Enright for encouraging me to continue the seminar during the period it was being run, and for help in checking the manuscript. Without this help these notes would not have appeared. I am above all deeply grateful to Harish-Chandra for giving me his time and ideas so generously during my various visits to Princeton and for helping me to understand his view of the subject.

Charlotte Johnson typed these notes with great skill, patience, and speed. I am very grateful to her for putting up with all my demands and carrying out the many and often confusing changes I wanted. Alice Hume typed an early draft of a section of these notes and Elaine Barth helped me in preparing these notes at all stages. To both of them my gratitude.

Finally, I wish to acknowledge my indebtedness to various institutions and Foundations that supported me during the many stages of the preparation: to the Alfred P Sloan Foundation; to the National Science Foundation for the grant that has supported me over the past several years; to the I.H.E.S. at Bures/Yvette, the Mathematics Institute of the Rijks University of Utrecht and the Institute for Advanced Study at Princeton for their hospitality during 1975; and, to the lively and diversified group of young men and women at the Huize Fatimah in Zeist, Holland for providing me with a most unusual working atmosphere during the Fall of 1975 when I wrote these notes in their present form.

4.2. Review by: Rebecca A Herb.
Bulletin of the American Mathematical Society (N.S.) 1 (2) (1977), 398-401.

The study of harmonic analysis on real semisimple Lie groups has proceeded in three major currents. One of these has been via the study of more general locally compact groups. There are, for example, general results about character theory, induced representations, and the Plancherel theorem. Good references for this work are the books of Dixmier and Mackey.

In the other extreme, there have been studies of specific groups, most frequently $SL(2, \mathbb{R})$, the group of $2 \times 2$ real matrices with determinant one. Although this is the simplest possible example of a semisimple real Lie group, studies of $SL(2, \mathbb{R})$, two of the earliest being papers by E Wigner in 1939 and V Bargmann in 1947, provided invaluable inspiration for the general theory.

Finally, there has been the study of real semisimple Lie groups in general. The pioneering work in this area is due almost entirely to Harish-Chandra. This work exploits the rich structure theory of semisimple groups and the connections between analysis on these groups and the abelian Fourier analysis on their Lie algebras and Cartan subgroups.
...
...
Experts should find Varadarajan's book a convenient reference for material which is scattered among many of Harish-Chandra's papers. Varadarajan follows Harish-Chandra's point of view closely. However, some of Harish-Chandra's proofs, for example, the construction of the discrete series and the proof that the orbital integral of a Schwartz function is again a Schwartz function are simplified significantly. Also, Varadarajan's detailed introductions provide an overview of the development which is very difficult to glean from Harish-Chandra's papers. Much of the analytic machinery is also available in Chapter 8 of G Warner's, Harmonic analysis on semi-simple Lie groups, vol. II. However Warner's book does not deal systematically with the reductive groups of class H or include details on the construction of the discrete series characters.

One minor obstacle to the book's usefulness as a reference is two different sets of page numbers. Also, although the list of symbols and index are detailed, in at least one case the page reference was incorrect.

An obvious limitation to Varadarajan's book is that it mentions only the work of Harish-Chandra. One problem that Harish-Chandra's work leaves open is the determination of certain integer constants which appear in formulas for the discrete series characters on regular elements not contained in any compact subgroup. These values are needed when the discrete series characters are used in specific problems of harmonic analysis such as Fourier inversion of orbital integrals. Progress has been made on this problem by T Hirai, and by W Schmid and his students, among others.

An important aspect of the discrete series not treated by Harish-Chandra is the problem of concrete realisation of the representations. Various geometric realisations, analogous to the Borel-Weil theorem for compact groups, have been established, inspired by a conjecture of Langlands. The complete problem of existence and realisation of the discrete series has recently been treated from a significantly different point of view by M Atiyah and W Schmid, and the work of Schmid has inspired many recent developments in quite a different spirit than the work of Harish-Chandra.

None of these developments are mentioned in Varadarajan's book. Although a complete treatment of this material would be outside the scope of Varadarajan's book, some remarks and a good list of references would have been useful in bringing it up to date on the theory of the discrete series.

However, recent developments do not diminish the importance of Harish-Chandra's original work. The analytic machinery used such as the Schwartz space and orbital integrals are of independent importance. In addition, Harish-Chandra's construction has been useful in providing an analogy for understanding the representation theory of reductive $p$-adic groups. Virtually all progress which has recently been made in the study of general reductive $p$-adic groups has followed Harish-Chandra's programme.

4.3. Review by: A U Klimyk.
Mathematical Reviews MR0473111 (57 #12789).

These notes are based on lectures that the author gave during 1969-1973 at the University of California at Los Angeles. The aim of them is to give a self-contained exposition of Harish-Chandra's work on harmonic analysis on real reductive groups, leading to the complete determination of the discrete series. The author remains quite close to Harish-Chandra's view of the subject, although there are some differences in details.

There are two parts. Part One deals with the problems of invariant analysis on a real reductive Lie algebra ...
...
Part Two is devoted to the explicit determination of the characters and formal degrees of the discrete series of representations. Here descent to the Lie algebra plays a key role in many proofs. ...
...
The appendices contain estimates of solutions of some ordinary differential equations and a study of certain representations of polynomial algebras associated with finite reflexion groups.

5.1. From the Preface.

You have only right to your actions, never to the fruits.

It was about four years ago that Springer-Verlag suggested that a revised edition in a single volume of my two-volume work may be worthwhile. I agreed enthusiastically but the project was delayed for many reasons, one of the most important of which was that I did not have at that time any clear idea as to how the revision was to be carried out. Eventually I decided to leave intact most of the original material, but make the current edition a little more up-to-date by adding, in the form of notes to the individual chapters, some recent references and occasional brief discussions of topics not treated in the original text. The only substantive change from the earlier work is in the treatment of projective geometry; Chapters II through V of the original Volume I have been condensed and streamlined into a single Chapter II. I wish to express my deep gratitude to Donald Babbitt for his generous advice that helped me in organising this revision, and to Springer-Verlag for their patience and understanding that went beyond what one has a right to expect from a publisher.

I suppose an author's feelings are always mixed when one of his books that is comparatively old is brought out once again. The progress of Science in our time is so explosive that a discovery is hardly made before it becomes obsolete; and yet, precisely because of this, it is essential to keep in sight the origins of things that are taken for granted, if only to lend some perspective to what we are trying to achieve. All I can say is that there are times when one should look back as well as forward, and that the ancient lines, part of which are quoted above still capture the spirit of my thoughts.

5.2. Review by: Robin L Hudson.
The Mathematical Gazette 70 (454) (1986), 332-333.

This fine book is a new edition of the author's work of the same title previously published in two volumes in 1968 and 1970. It differs from the first edition principally in that three of the original chapters on projective geometry are now condensed into a single one, but many references to modern work on foundations of quantum theory have also been inserted. The book has worn well and the new edition is welcome. Prospective readers should take note of the author's assertion, in the preface to the first edition, that "the reader with a knowledge of abstract integration, Hilbert space theory and topological groups will find the going easy".

The book is designed for the reader who can supplement a desire to understand quantum theory with a degree of mathematical sophistication.

The central mystery of quantum theory is its linearity: why are quantum states described by vectors in a linear space and observable quantities by linear operators upon them? The approach to the unfolding of this mystery followed here is based on generalisations of the fundamental theorem of projective geometry. Namely, for a given abstract projective geometry comprising objects such as points, lines, planes, etc, satisfying certain axioms such as that every pair of points lie on a unique line, every pair of coplanar lines meet in a unique point and so on, it is always possible to introduce a coordinate system, or equivalently to find a vector space such that the points correspond to one-dimensional subspaces, the lines to two dimensional subspaces and so on. By postulating axioms for the experimentally verifiable propositions concerning a physical system which resemble those of a generalised projective geometry, one is thus led in an intuitively satisfying way to linear space structures. However because the underlying number field of the vector space is manufactured out of the geometry itself, it is in general unrelated to the real or complex numbers, and to introduce metric or analytic constructions and the familiar Hilbert spaces of quantum theory requires further delicate considerations.

Having arrived at the "correct" linear space model for the propositions, one may now understand why observables should be representable by operators; the key here is the spectral theorem for a self-adjoint operator in a Hilbert space and its converse, which establish a one-one correspondence between such operators and projection-valued measures on the real line; the latter being in a natural way identified with the observables. Similarly, the celebrated and notoriously difficult Gleason's theorem, of which a full proof is given in this book, shows that states, defined naturally as probabilities on the propositions, are indeed characterised by unit vectors in the underlying Hilbert spaces, the familiar wave functions of quantum theory, in the "pure" case, and are in general mixtures of states characterised in this way.

A third, and surpassingly beautiful, understanding comes about from the implementation of symmetry within this scheme; symmetries correspond to automorphisms of the proposition structure, and as such are related through another famous theorem of Wigner to automorphisms of the underlying Hilbert space, and of its underlying (real, complex or quaternionic) number field. Thus we are led to the author's amour propre, the representation theory of Lie groups, and of the Lorentz and Galileian groups in particular, and to the classification of these representations by means of parameters interpreted physically as the mass and spin of elementary particles, a theory whose success is the inspiration of more modern particle theories based on representations of other groups.

This is not an easy book to read; not only is the subject material profoundly deep, but the language used is austere and uncompromising. The rewards for doing so in understanding and enlightenment are correspondingly immense, and the potential reader should not be misled by the author's initial quotation from the Bhagavadgita which can be translated as: "dedicate thyself only to thy labours, never to the fruits thereof."

5.3. Review by: Stanley P Gudder.
Mathematical Reviews MR0805158 (87a:81009).

This book is a one-volume second edition of the original two-volume edition with the same title [1.1 and 1.2 above]. The two volumes of the first edition have greatly influenced researchers in this field and are now considered classics. Most of the original material has been retained and the current edition is updated with the addition of notes to the individual chapters, recent references and occasional brief discussions. The notes at the end of Chapter IV are particularly lucid and give the reader an added perspective in placing the book's material into the context of current research. These notes discuss hidden variables, the algebraic formulation of quantum mechanics, statistics of mixed states, composite systems, measurement in quantum mechanics, quantum probability theory, and generalisations of Gleason's theorem to von Neumann algebras. The main change in the current edition is that Chapters II through V of the original Volume I have been condensed into a single Chapter II.

The book has three main parts. The first part introduces the viewpoint of von Neumann in which every physical system can be described in terms of an orthocomplemented lattice (called the logic of the system) whose elements represent experimentally verifiable propositions about the system. The main result in this part is the fundamental theorem of projective geometry. The second part analyses the structure of the logics of quantum systems. Results concerning simultaneous observability are given and this part concludes with Piron's representation theorem, which proves that projective logics are precisely the standard Hilbert space logics.

The third part deals with specialised questions. The main concern is that of a covariant description of a quantum system, the covariance being with respect to suitable symmetry groups of the system. The Schrödinger and Dirac equations are obtained and the classification of a single-particle system is derived in terms of its mass and spin.

It is the author's view that the geometric approach is the clearest and smoothest path to the fundamental results of the quantum mechanical world.

6.1. From the Preface.

Analysis on Symmetric spaces, or more generally, on homogeneous spaces of semisimple Lie groups, is a subject that has undergone a vigorous development in recent years, and has become a central part of contemporary mathematics. This is only to be expected, since homogeneous spaces and group representations arise naturally in diverse contexts ranging from Number theory and Geometry to Particle Physics and Polymer Chemistry. Its explosive growth sometimes makes it difficult to realise that it is actually relatively young as mathematical theories go. The early ideas in the subject (as is the case with many others) go back to Elie Cartan and Hermann Weyl who studied the compact symmetric spaces in the 1930's. However its full development did not begin until the 1950's when Gel'fand and Harish-Chandra dared to dream of a theory of representations that included all semisimple Lie groups. Harish-Chandra's theory of spherical functions was essentially complete in the late 1950's, and was to prove to be the forerunner of his monumental work on harmonic analysis on reductive groups that has inspired a whole generation of mathematicians. It is the harmonic analysis of spherical functions on symmetric spaces, that is at the focus of this book.

The fundamental questions of harmonic analysis on symmetric spaces involve an interplay of the geometric, analytical, and algebraic aspects of these spaces. They have therefore attracted a great deal of attention, and there have been many excellent expositions of the themes that are characteristic of this subject. We would like to mention for instance the beautiful books of Helgason, especially for their treatment of the differential geometric side of the theory. However there has been, as yet, no detailed account of the spectral theory of symmetric spaces. Our aim in this book is to give a self-contained account of this aspect of spherical function theory. While it is true that the harmonic analysis of spherical functions on symmetric spaces may be subsumed under the more general theory of Harish-Chandra that deals with Eisenstein integrals, the latter is technically very much more involved because it is tied up with many difficult themes in representation theory of reductive groups such as the discrete series, geometry of parabolic subgroups, analysis of unstable differential equations at infinity, and so on. It is our firm belief that the spherical case offers many deep glimpses into the general situation, and that one can get a great deal of insight into the spectral theory of Eisenstein integrals from a study of the spherical case.

We have left off many interesting aspects of these functions in this book, such as, non LP-theory, the theory of boundaries, the harmonic analysis of spherical functions on $p$-adic reductive groups, to mention a few. We feel however that this has resulted in sharpening our focus considerably.

It is a pleasure to acknowledge the debts that we have been allowed to accumulate during the writing of this book. We have received much kindness and help from many friends. We are deeply grateful to the editors of the Ergebnisse, and especially to Peter Hilton, for their initial invitation to write this monograph, and even more, for their great patience in bearing with us while the project suffered many delays stemming from a variety of personal and professional reasons. Among many friends who made comments, corrected our blunders, and rendered general advice, we like to mention Tom Enright, Sig Helgason, Joop Kolk, Hans Duistermatt, Brad Shelton, Alladi Sitaram, Peter Trombi, and Garth Warner. One of us (V) wants to acknowledge the hospitality of the State University of Utrecht at Utrecht and the Tata Institute of Fundamental Research at Bombay for inviting him to spend substantial amounts of time there; large parts of the material in this book were covered in lectures given in these places. Pat Monohon and Veda Varadarajan helped us with the typing and the corrections; no simple thanks are sufficient for their skill and patience.

Above everything stands our debt to Harish-Chandra. His influence pervades the entire book, not only because he is the central figure in the subject, but also due to the fact that our contacts with him have shaped our mathematical thinking in a profound manner. We shall always remember him as a friend and a teacher, and we respectfully dedicate this book to his memory.

We would like to conclude this preface on a personal note. The sudden and untimely passing away of Walter Kaufmann-Bühler, Mathematics Editor of Springer-Verlag, just before the final preparation of this book, was a great shock to all his friends and colleagues. We shall always cherish his unfailing courtesy, good humour, and generosity of spirit, as well as the help and encouragement he gave us throughout the preparation of this book. The warmth of our appreciation is enveloped by the sadness that attends the loss of a very good friend.

6.2. Review by: D Barbasch.
Bulletin of the American Mathematical Society (N.S.) 23 (2) (1990), 598-604.

The book by Gangolli and Varadarajan gives an account of the theory for semisimple symmetric spaces. The core of the book exposes ideas in the work of Harish-Chandra on asymptotics of elementary spherical functions viewed as eigenfunctions ....

Chapter 1 gives an overview of the abstract Plancherel formula for the case of spherical functions. Chapters 2 and 3 give the basic more elementary results needed for Chapters 4 and 5. Chapter 6 contains additional results on $L^{p}$ with $p ≠ 2$.

The style of the book is very clear and essentially complete details are given, except for the expository parts, where references are cited. Ample discussions after each chapter trace the history of the subject starting with the work of Gel'fand and Harish-Chandra in the early fifties and continuing with the contributions of Kostant, Helgason, the two authors themselves, and many others. It is an excellent survey of the theory of spherical functions for Riemannian symmetric spaces.

As far as considering it as a textbook for a graduate course, I think it would only be suitable for an advanced course, possibly a reading course only. For one thing, the techniques used come from many different fields - partial differential equations, differential geometry, Lie groups and Lie algebras, functional analysis, and so on. In this respect, I would highly recommend Chapters 1 and 2 for the expository survey of representation theory, spectral theory of Banach algebras, and just basic structure of Lie groups. In a subject as technical as this, it is important not to lose sight of the main ideas.

6.3. Review by: Roberto J Miatello.
Mathematical Reviews MR0954385 (89m:22015).

Let G be a real reductive group in the Harish-Chandra class. Let $K$ be a maximal compact subgroup, $G=KAN$ a fixed Iwasawa decomposition of $G$, ... . The present book is a comprehensive treatment of the theory of (class 1) $K$-spherical functions on $G$. An outline follows.

Chapter 1 includes, in a general setting, a review of results in representation theory and an exposition of the basic properties of spherical functions (with respect to an arbitrary $K$-type). In the last two sections, an abstract Plancherel formula is proved for $(G, K)$ a Gelʹfand pair, and the connection with eigenfunction expansions of invariant differential operators in $G/K$ is discussed.

Chapter 2 is devoted mainly to the structure theory of the groups of Harish-Chandra class (Iwasawa and Bruhat decompositions, parabolic subgroups and subalgebras). Also, various integral formulas are given and the Harish-Chandra homomorphism is discussed in detail.

Chapter 3 includes a discussion on the minimal principal series representations, the determination of all elementary spherical functions on $G$ and the definition and the most basic properties of the Harish-Chandra transform. ...
...

7.1. From the Preface.

A substantial part of the material covered in these notes formed the content of a course of lectures given during the Spring of 1985 in the Mathematics Institute of the University of Warwick, England. My aim was to introduce the aspiring graduate student to a beautiful and central part of mathematics, the representation theory of semisimple groups. This is of course a vast and active subject, bringing together at a fairly deep level algebra, geometry, analysis, and arithmetic. This is one of the reasons why it is difficult to get into, at least for the young student. I therefore made an attempt to keep the requirements minimal, and introduced the major themes of the subject by first working them out in the case of SL(2, R). This approach has no claim to novelty: it has been done before, and there is the well-known book of S. Lang dealing only with $SL(2, \mathbb{R})$. I have, however, discussed a number of topics not treated by Lang, such as the Schwartz space, invariant eigendistributions, wave packets, and so on; in addition, I have included, wherever possible, indications of how these ideas may be generalized to the context of a general semisimple Lie group.

The organisation of the book is not always linear because I wanted to adhere closely to the lectures and preserve their free-flowing nature. As a result, the reader will often find references to matters that are not defined or are quite advanced, especially in Chapters 1-2. This should not discourage him (or her); my advice to the reader is to ignore them and proceed ahead, and come back to the difficult points later. Chapters 4-8 are essentially linear and can be worked through by a graduate student (late first year or early second year in an American university). I have included appendices on Functional Analysis and Lie theory that offer the reader some basic definitions, explanations of some concepts, and some historical perspective.

It is a great pleasure for me to express my gratitude to the Mathematics Institute of the University of Warwick and the Science and Engineering Research Council of the United Kingdom for inviting me to Warwick, and to the staff and faculty of the Institute for their wonderful hospitality. To thank Klaus and Annelise Schmidt adequately for what they did for my wife and me is impossible; it was only through their kindness and generosity that this visit become so memorable for us. Finally I want to thank the Cambridge University Press and their editor Dr David Tranah for being really patient while I took my time preparing the final version of this manuscript.

7.2. Review by: Joseph A Wolf.
Bulletin of the American Mathematical Society (N.S.) 28 (2) (1993), 367-370.

Semisimple Lie groups are symmetry groups that occur in surprisingly many situations. They are the isometry groups of Riemannian symmetric spaces, the analytic automorphism groups of bounded symmetric domains, the groups from which Eisenstein series and cusp forms are constructed in analytic number theory, the conformal groups of general relativity, the groups whose representations correspond to elementary particles, .... They should form part of the basic toolkit of every modern mathematician; but, in fact, the theory is relatively unknown because it is not easily accessible.

The reason for the inaccessibility of semisimple Lie group theory is clear to anyone who has tried to learn or to teach it: One must navigate a path too complicated to follow without a good vehicle and a good map. But it's well worth the effort: that path leads to a breathtaking mathematical vista.

Varadarajan's book Harmonic analysis on semisimple Lie groups is the best introduction to harmonic analysis on semisimple Lie groups from the analytic viewpoint. It is neither a textbook nor a monograph in the usual sense, but rather a sort of pedagogic discourse that exposes the reader to semisimple Lie theory in a useful and informative way. After reading this book, one can either stop with a pretty good understanding of the theory and its role in harmonic analysis (if not in geometry, probability, or physics), or one can continue to study the theory with a reasonable background and an excellent sense of direction. Also, and this is no small matter, the book is a pleasure to read.

There are other important viewpoints for harmonic analysis on semisimple Lie groups and their homogeneous spaces. One has a viewpoint oriented toward linear algebraic groups that includes the theory of $p$-adic semisimple groups, a viewpoint oriented toward number theory that includes automorphic representation theory, a viewpoint of Riemannian geometry and symmetric spaces, a viewpoint of particle physics, and their various mixtures. But life is too short to discuss those here.

Varadarajan's book begins with an interesting introduction to harmonic analysis, with reference to classical Fourier analysis and several accessible applications. Then the book starts in a serious way with a quick sketch of harmonic analysis on compact groups. The Peter-Weyl Theorem, which extends the method of harmonic analysis from Fourier development of periodic functions of one variable to functions on general compact topological groups, is described with emphasis on representations of the group. Weyl's character formula and the Plancherel formula, in terms of Cartan's highest weight theory, are indicated for the unitary group. These methods, and the connection between compact and complex groups, then are briefly indicated in general. This is followed by a concise sketch of some basic facts on unitary representations and harmonic analysis for locally compact groups. Varadarajan gives a brief history, says some words about Haar measures on groups and quotient spaces, sketches the theory of induced representations, says a few words about factor representations, and concludes:

This is the historical background of our subject. its thrust is very clear; classical Fourier analysis is a very uncertain guide in predicting how noncommutative harmonic analysis should be done. It is thus not surprising that already in the middle and late 1940s some people were turning their attention to some of the simplest noncompact nonabelian groups like $SL(2, \mathbb{R})$ and $SL(2, \mathbb{C})$, to get a better insight into the new phenomena which arise. The work of Gel'fand-Naimark, Bargmann, and Harish-Chandra dates to this period and marks, in my opinion, the real beginning of modern noncommutative Fourier analysis and representation theory.

Every mathematician should know the material in the three chapters just described.
...
...

7.3. Review by: Takeshi Kawazoe.
Mathematical Reviews MR1071183 (91m:22018).

Harmonic analysis on semisimple Lie groups G, essentially developed by Harish-Chandra in the years 1950-1980, is one of the most profound and beautiful achievements in the mathematics of our time. One of the objectives is the decomposition theory of the regular representation of $G$, and the Plancherel formula for the Schwartz space $<curlyC> (G)$ is a sufficient goal. The aim of this book is to introduce advanced undergraduates and beginning graduate students to this subject and a number of related topics such as characters, invariant eigendistributions, wave packets, Harish-Chandra transform, and so on. As many books do, it is effective and essential to introduce such vast subjects by working them out in the case of $SL(2, \mathbb{R})$, in order to minimize the requirements. However, sometimes explicit calculation tells us nothing about what is likely to happen in the general case. In this book the author develops faithfully in $SL(2, \mathbb{R})$ what Harish-Chandra carried out in the general case, so we can well understand his programme and ideas.

In Chapters 1 and 2 the representation of groups and Fourier analysis on abelian and compact groups are explained. Although some parts of the contents such as invariant theory, quantum mechanics and arithmetic are far advanced for the beginner, the reviewer feels that representation theory ranges far and wide. The representation theory and Fourier analysis on groups neither compact nor abelian start at Chapter 3 and are developed to a final goal in Chapter 8 - the characterisation of Harish-Chandra transforms of (G) and transforms of tempered invariant distributions on G. The proof of the latter was worked out by R Ranga Raoand the author. ...
...

8.1. From the Preface.

This book arose out of a course entitled

Mathematics in ancient and modern times: The birth of modern algebra

which I gave to a small group of freshmen students in UCLA in the Fall of 1993. The students had only high school mathematics as their background and so I did not use calculus. However I assisted them so that they selected individual projects to work on as their assignment for fulfilling the requirements of the course. Each project was designed to have two parts: a historical and a mathematical one. The projects had wide variety which surprised me, but which increased my pleasure as the course progressed. This was the first time that I had taught a small group of freshmen in a format different from the usual cook-book type in more than thirty years of teaching in UCLA, and the experience was memorable.

My inspiration for designing and teaching such a course came after reading the beautiful book Tales of physicists and mathematicians by my friend Simon Gindikin, especially his account of the life and work of Cardano and its influence on the rise of modern algebra. However in the course I did not confine myself to the work of Cardano but explored a wider theme, namely the growth of algebra from its ancient origins to its current state that allows us to have a marvellous understanding of the whole gamut of physical sciences: from the abstract reaches of number theory to the wonders of quantum physics.

In 1996 Rajendra Bhatia of the Indian Statistical Institute suggested the possibility of making a book out of these notes. I then revised the notes adding exercises and additional material-partly historical and partly dealing with more advanced topics in the form of notes and exercises at the end of the individual chapters. These may also be used as starting points for projects for the more ambitious student. The result is what I hope is an interesting way to present the basics of modern algebra at a very elementary level. The approach that I have adopted here (as in the course) is what Shafarevich calls the biogenetic one; the student is exposed to the historical and evolutionary development of the subject as an essential part of the course, and the material is presented in a way that allows him or her to see clearly how ideas have emerged in the attempt to solve specific questions.

I am not sure how successful my attempt has been. However it was a great experience for me, not only at the time the course was given, but also later when the book was written. I would like to record my gratefulness to the Honors Collegium of UCLA which encouraged me to give the course, to the small group of young men and women who took the original course, and to R Bhatia who encouraged me to enlarge the notes into its present form. Finally, I would like to thank Wissam Barakat for his great technical help in the preparation of the final version of the manuscript.

8.2. Review by: James V Rauff.
The Mathematics Teacher 92 (5) (1999), 454.

This book is a personal, rather than comprehensive, look at the development of algebra. It emphasises Indian contributions and great developments in Italy in the thirteenth and fourteenth centuries. The treatment of modern algebra is idiosyncratic, and most readers will find unexpected omissions; for example, Noether is not mentioned. The book would probably not be appropriate as a textbook for a general history-of-mathematics course but would serve admirably as one component of such a course. Nevertheless, Varadarajan spins a captivating tale, and the mathematics is first-rate. The book belongs on the shelf of any teacher of algebra.

The author structured the book around the quest for closed-form solutions of polynomial equations. As a result, readers are treated to detailed expositions of Scipione del Ferro's solution to $X^{3}+ PX = Q$, Cardano's generalisation of Tartaglia's solution to the cubic equation, the dispute between Cardano and Tartaglia, and Ferrari's solution to the quartic equation. The impossibility of solving all quintic equations in terms of radicals is mentioned, but no details are given. Modern algebra and the axiomatic method are introduced through Hamilton's quaternions.

The discussion of European algebra is well written but unexceptional. The great treasure of this book is the discussion of the work of the great Hindu mathematicians Aryabhata (c. 476-550), Brahmagupta (c. 598-665), and Bhaskara (c. 1114-1185). Teachers of mathematics history will be especially interested in Varadarajan's exposition of the remarkable cakravala, an algorithm for solving $X^{2}- NY^{2} = ±1$.

The book contains many exercises that enhance and supplement the text and that also include historical information. Many of the exercises ask readers to apply the historical techniques. Some of the exercises are quite difficult and will challenge any student.

8.3. Review by: Victor J Katz.
Mathematical Reviews MR1621032 (99d:01007).

This work is a brief survey of aspects of the history of algebra from Babylonian times to the twentieth century, as presented by the author in a course at UCLA. There is little new in the book, but Varadarajan gives us nice treatments of the work of Indian mathematicians on the so-called Pell equation as well as a very detailed yet teachable discussion of the standard story of the solution of cubic and quartic equations by del Ferro, Tartaglia, Cardano, and Ferrari in sixteenth-century Italy. Unfortunately, the author is evidently not aware of the latest scholarship on such topics as the Babylonian tablet Plimpton 322 and its Pythagorean triplets, or either the medieval Islamic or 14th century Italian work on cubic equations. The final section of the book gives a modern treatment of such topics as complex numbers and the fundamental theorem of algebra, in which the author explains more fully the solution of cubic equations.

9.1. From the Introduction.

It was Professor S T Yau and Professor S Y Cheng who first suggested that the International Press bring out a volume of my Selected Papers. Since then it has turned out that this would be done by the American Mathematical Society in collaboration with the International Press, and this volume is the result. It is a signal honour for me and I am grateful to everyone who made it possible.

It is with very mixed feelings that one looks back at what one has accomplished. There is pleasure of course but there is also a feeling of humility since one knows what one's dreams were and what a minute fraction of them were actually achieved. That some part of the dreams were made real at all is due to my various collaborators. Not only do I regard myself as fortunate in having worked with them, but I have also no doubt that their resolve and strength were critical for pushing through my vague projects which were, most of the time, beyond my capacities to carry out by myself.

I wish to pay my respects at this time to those I regard as my teachers in the sense of having shaped, by their advice and their willingness to share their insights, my mathematical personality - R Arens, V Bargmann, P Deligne, R J Finkelstein, Harish-Chandra, G W Mackey, C R Rao and R Steinberg.

Above everything I am grateful to Enrico Beltrametti, Gianni Cassinelli, and Piero Truini for reviving my interest in physics and showing always an extraordinary solicitude for me; to D G Babbitt, S D Chatterji, J A C Kolk, K R Parthasarathy, and S R S Varadhan for a lifetime of friendship; and to my wife Veda, not only for sharing my life but also for her wise counsel that made sure that I was not lost among inconsequential things.

9.2. Review by: Rebecca A Herb.
Mathematical Reviews MR1647109 (99j:01030).

This book is a collection of 22 papers by V S Varadarajan. In addition to the papers themselves, the volume contains a complete bibliography of his 64 papers and 6 books published between 1958 and 1998. Finally, it has a very useful set of comments by Varadarajan that serve as a historical introduction to the papers included in the volume. The papers cover a wide variety of topics. The early papers study measures on topological spaces with an emphasis on probabilistic aspects and applications to quantum mechanics. The second group of papers deals with the representation theory of semisimple Lie groups. This section includes important papers from his collaborations with Trombi, Enright, and Duistermaat and Kolk. The next group of papers describes Varadarajan's work with Babbitt on the structure of ordinary differential equations in the complex domain. The final group of papers studies aspects of representation theory of interest in physics, and includes the study of certain non-unitary representations of the Poincare group, joint work with Truini on quantum groups, and work on quantum systems. In order to survey Varadarajan's work over a period of forty years adequately in one volume, the editors have been forced to omit many of the important longer papers. However, they have included a number of expository articles which explain the ideas and techniques of these longer papers. Varadarajan's historical comments on each topic together with the bibliography will make it easy for the interested reader to refer back to the original papers.

10.1. From the Preface.

See 7.1. above.

10.2. Review by: Krishnan Parthasarathy.
Mathematical Reviews MR1725738 (2000m:22011).

Semisimple Lie groups have some kind of ubiquity, arising as they do in such diverse areas as analysis, Riemannian geometry, number theory, general relativity and particle physics. Infinite-dimensional representation theory and analysis on semisimple groups have come to occupy the centre stage in contemporary mathematics. This pre-eminence of semisimple groups in mathematics is largely due to the extraordinary and epoch-making work of Harish-Chandra. With a vision that ordinary mortals can only wonder at for its depth and clarity, he set his goal and worked towards it systematically and relentlessly with amazingly intense single-mindedness of purpose. This unique pursuit, spanning almost three decades and an astounding sequence of papers of great depth, marking a profound synthesis of algebra, analysis and geometry, resulted in the erection, almost single-handedly, of a monumental theory of harmonic analysis on semisimple groups. The great power of his methods and ideas continues to reverberate in terra mathematica. His achievement is, without doubt, one of the greatest in mathematics of the twentieth century.

Turning his attention to $p$-adic groups, he applied his methods to obtain the Plancherel formula for these groups. (The problem of discrete series, however, is not yet completely settled.) The influence of his powerful methods and constructions on the theory of automorphic forms is pervasive. His work forms the analytical foundation of the famous Langlands programme (which, in the words of J. Arthur, "represents a profound organising scheme for fundamental arithmetic data in terms of highly structured analytic data"). This programme has greatly influenced progress in arithmetic in the last few decades. A stunning illustration is Andrew Wiles's spectacular proof of the semistable case of the Shimura-Weil-Taniyama conjecture and the consequent establishment of Fermat's last theorem. (Incidentally, a proof of the general form of this conjecture has been announced very recently.)

Varadarajan, a leading expert in the subject, seeks to introduce the prospective reader of his book to Harish-Chandra's fascinating work on real semisimple groups by looking at the particular group $SL(2, \mathbb{R})$. The important point, however, is that although the details are worked out only in the special case, the Harish-Chandra perspective is kept in focus throughout. In fact, the book may be described as a long, illuminating discourse on Harish-Chandra's ideas, methods and philosophy, together with illustrations of these in the particular case.

The introductory chapter serves as an apéritif for the gourmet dinner served later. It provides a rapid view of the picture in the classical abelian case and offers little nuggets on sources of motivation and on some applications. The second chapter contains quick, nice fare on compact groups à la Hermann Weyl offering all the items of a standard menu: orthogonality relations, Peter-Weyl theorem, Weyl's character formula, Weyl's unitary trick, etc. The infinitesimal approach, studying the Lie group via the Lie algebra, originally pioneered by Élie Cartan and later given a conceptualised treatment by Chevalley and Harish-Chandra, is discussed and the highest weight theory is presented. Orbital integrals and their relation to the Plancherel theorem are studied, as a precursor for the later discussion on noncompact groups. A brief look at general representation theory, including induced representations, and a miniature historical picture form the content of Chapter 3.

The remaining chapters, treating harmonic analysis on noncompact semisimple groups, are more difficult and much more rewarding. The method of constructing irreducible unitary representations of complex groups as induced representations from parabolic subgroups, pioneered by Gelʹfand and Naĭmark, is expounded and relevant character calculations are made in Chapter 4. The stage is also set for the Harish-Chandra perspective in the real case: restricted parabolic induction and construction of discrete series.

Shuttling between "infinitesimal" and "global" pictures was not difficult in the case of finite-dimensional representations. Harish-Chandra pioneered the infinite-dimensional case in a remarkable series of papers. His approach is discussed in Chapter 5. Analytic vectors, the Harish-Chandra density theorem and Harish-Chandra modules are among the central pieces in this chapter, which concludes with some explicit calculations for $SL(2, \mathbb{R})$ and $SL(2, \mathbb{C})$.

The Plancherel formula for $SL(n, \mathbb{C})$ was derived by Gelʹfand and Naĭmark by applying Fourier transform theory to the limit formula for orbital integrals. Harish-Chandra's method of doing this for all complex semisimple groups heralds Chapter 6. This is followed by Harish-Chandra's powerful theory of orbital integrals in the real case, carried out for $SL(2, \mathbb{R})$. Based on Harish-Chandra's regularity theorem (discussed in the next chapter), the discrete series for $SL(2, \mathbb{R})$ is constructed and the Plancherel formula is obtained.

Harish-Chandra realised early that, unlike in the complex case where discrete series did not exist, the construction of discrete series held the key to success in the real case. This construction became the focus of his prodigious efforts. This Herculean achievement was an illustration of André Weil's remark that for Harish-Chandra "technical difficulties simply did not exist". This great work was based on his explicit construction of invariant eigendistributions parametrised by the characters of a compact Cartan subgroup. A sketch of this deep theory for $SL(2, \mathbb{R})$ is given in Chapter 7. The pièce de résistance is, of course, the great regularity theorem. This is a deep result and is (in words that Varadarajan uses in this context elsewhere) the magic "key that opens all the doors" for a smooth flow of the theory. It is one of Harish-Chandra's greatest achievements and is a veritable tour de force. Varadarajan shows how Harish-Chandra's work on discrete series may be viewed as a profound generalisation of Weyl's character formula in the compact case. The author concludes the chapter with the following words: "If one wants to extend all of this to general semisimple groups, it is clear what one must do. The orbital integrals, radial components and distributions with unipotent support must all be studied in complete generality. That, however, will take anyone along Harish-Chandra's epic journey."

The longest chapter of the book is the last one, on the analysis of the Schwartz space. For SL(2, R) the Schwartz space was defined and studied originally by Ehrenpreis and Mautner. But, in keeping with the spirit of the book, Varadarajan gives the general method of Harish-Chandra specialized to the case of $SL(2, \mathbb{R})$. Excellent discussions on wave packets, the Harish-Chandra transform, tempered distributions, the Plancherel formula in the Schwartz space and the completeness theorem for the Harish-Chandra transform are all given here, showing that the Schwartz space provides the correct framework for $L^{2}$-harmonic analysis.

Three short appendices on functional analysis, topological groups and Lie groups and Lie algebras conclude the book.

Varadarajan, the expert guide, leads the willing traveller on an unforgettable journey through the fascinating contours of the Harish-Chandra country, revealing countless vistas of great beauty and pointing out high peaks, at once lofty and austere. For a novice, uninitiated in the intricacies of Lie theory (the reviewer is not much better off), trying to learn harmonic analysis on semisimple groups, there can be no better place to start. Anyone who has completed this "mini journey" with diligence can confidently undertake to follow Harish-Chandra's "epic journey" from more advanced monographs and Harish-Chandra's original papers.

The author deserves gratitude for writing such a beautiful book for the beginner. The less expensive paperback in this corrected printing of the book is welcome. The reviewer feels honoured on being asked to review this book and is doubly happy to get a copy of the book as a bonus - a case of getting paid to be taken on a wonderful tour!

11.1. From the Preface.

These notes are essentially the contents of a minicourse I gave at the Courant Institute in the fall of 2002. I have expanded the lectures by discussing spinors at greater length and by including treatments of integration theory and the local Frobenius theorem, but otherwise have not altered the plan of the course. My aim was and is to give an introduction to some of the mathematical aspects of supersymmetry with occasional physical motivation. I do not discuss supergravity.

Not much is original in these notes. I have drawn freely and heavily from the beautiful exposition of P Deligne and J Morgan, which is part of the AMS volumes on quantum field theory and strings for mathematicians, and from the books and articles of D S Freed and D A Leites, all of which and more are referred to in the introduction.

I have profited greatly from the lectures that Professor S Ferrara gave at UCLA as well as from many extended conversations with him, both at UCLA and at CERN, where I spent a month in 2001. He introduced me to this part of mathematical physics and was a guide and participant on a seminar on supersymmetry that I ran in UCLA in 2000 with Rita Fioresi. I am deeply grateful to him for his unfailing patience and courtesy. I also gave a course in UCLA and a miniworkshop on supersymmetry in 2000 in Genoa, Italy, in the Istituto Nazionale di Fisica Nucleare. I am very grateful to Professors E Beltrametti and G Cassinelli, who arranged that visit; to Paolo Aniello, who made notes of my UCLA course; to Ernesto De Vito and Alberto Levrero, whose enthusiasm and energy made the Genoa workshop so memorable; and finally to Lauren Caston, who participated in the Courant course with great energy and enthusiasm. I also wish to thank Alessandro Toigo and Claudio Carmeli of INFN, Genoa, who worked through the entire manuscript and furnished me with a list of errors and misprints in the original version of the notes, and whose infectious enthusiasm lifted my spirits in the last stages of this work. I am very grateful to Julie Honig for her help during all stages of this work. Last, but not least, I wish to record my special thanks to Paul Monsour and Reeva Goldsmith whose tremendous effort in preparing and editing the manuscript has made this book enormously better than what it was when I sent it to them.

The course in the Courant Institute was given at the suggestion of Professor S R S Varadhan. My visit came at a time of mourning and tragedy for him in the aftermath of the 9/11 catastrophe, and I do not know how he found the time and energy to take care of all of us. It was a very special time for us and for him, and in my mind this course and these notes will always appear as a small effort on my part to alleviate the pain and grief by thinking about some beautiful things that are far bigger than ourselves.

11.2. Review by: Ben Lewis Cox.
Mathematical Reviews MR2069561 (2005g:58011).

The area of physics called supersymmetry describes an extension of the Standard Model of elementary particle physics that began in the early 1970's with the work of Y A Golʹfand, E P Likhtman, D V Volkov, V P Akulov, J Wess and B Zumino. By just skimming through the literature one realises that there is a whole industry of physicists working on the implications of supersymmetry. There are a number of reasons for which an elementary particle physicist would want to be knowledgeable about this rather vast topic. In addition to the text under review, B Greene's eloquent book [The elegant universe, 1999] has a superb description of this motivation. One timely reason for its study is that superpartners of bosons and fermions are expected to be detected in the new LHC (Large Hadron Collider) at CERN, which will be operational sometime in 2005 or 2006. Besides a few books and the twenty thousand plus articles written by physicists on supersymmetry, there are some mathematicians of note who have produced interesting work exploring the algebraic and geometric ramifications of the mathematical side of this subject.

The book under review has as its intended audience mathematicians with some graduate school education in Lie theory and differential geometry. Otherwise the book is self-contained. The introduction begins with a little prehistory of the subject, covering motivational material from classical and quantum mechanics, representation theory of the Poincaré group, Maxwell's equation for a photon, Dirac's equation describing relativistic particles, and Weyl's equation for a neutrino, and culminating in a short review of bosons and fermions. Since much of this might be new material for a mathematically oriented reader, Varadarajan does an excellent job conveying the spirit and background of the subject to someone with little knowledge of physics.

Varadarajan's broad knowledge of mathematics and its history shines through in the second chapter, which gives a leisurely review of the prehistory of the concept of a supermanifold. Here one finds a discussion of Riemann's prophetic vision of geometry. This definitely merits reading by geometers, string theorists and quantum field theorists, as the Riemannian viewpoint seems to be easily overlooked. Certainly Grothendieck's notion of a scheme was also a profound development in the field of algebraic geometry, unifying work in geometry, number theory, topology and complex analysis. ...
...
The book ends with a chapter on super Lie groups, the Poincaré-Birkhoff-Witt theorem for super Lie algebras and the classification of simple and classical super Lie algebras over an algebraically closed field as given by V Kac. The proof of the latter result is, however, not given.

Overall the book is well motivated with many clear explanations of the ideas used in the study of supersymmetry. I would recommend it to any mathematician or physicists interested in learning about this topic. The only modification that comes to mind would be for an index, as it would be useful for some one using this monograph as a textbook.

12.1. From the Preface.

The origins of this book go back to a course on the history of mathematics that I gave at UCLA in the winter quarter of 2001. In most universities such a course follows a standard curriculum that starts with the Babylonians and works its way to more recent topics. I decided to do it differently and focus attention on the work of a single great figure. I chose Euler, attracted both by his universality and the great relevance of his work to what should be, if not what is, the undergraduate program in mathematics today. I discussed mostly Euler's work on infinite series and products, and was guided by the chapter on Euler in A Weil's beautiful book Number Theory: An Approach through History from Hammurapi to Legendre that had been published in 1984. The result, as far as I was concerned, was predictable. I fell under the magical spell of Euler's personality and mathematics. It was a huge personal discovery for me to learn how alive Euler's themes still are. I started writing this book after the course, having been encouraged by my friends that it would be a good thing to try to do.

Before I get to Euler I really want to emphasise an important point. I believe that writing historically about mathematics should not be limited to questions like who did what when and to whom he/she wrote about it. I am of the opinion that the history of mathematics and mathematicians should go beyond these concerns, however legitimate they are. I feel that a complete mathematical history should pay great attention to the historical evolution of ideas and how they mesh with what we know and are interested in today. To modify a famous remark, the history of mathematics is too important to be left entirely to historians. No one has done this type of historical writing more brilliantly than A. Weil, as one can tell by his historical memoirs that are scattered throughout his Oeuvres Scientifiques. I have tried to follow his example: how far I have succeeded can be judged only by the readers.

No single person or book can describe completely the many-sided genius of Euler or his sunny and equable temperament that informs it. I have thus tried to limit my focus. I have been concerned only with the task of telling what the themes of Euler were and how they can be connected to current interests. Moreover, I have limited myself mostly to his work on infinite series and products and its repercussions in modern times, namely the theory of zeta values, and divergent series and integrals (Chapters 3 and 5). In Chapter 2 I have given a brief overview of some other parts of his work, for instance in elliptic integrals and number theory. His work on elliptic integrals is the forerunner of the modern theory of elliptic curves and abelian varieties, and his work on number theory raised questions which could only be understood fully after the development of class field theory. In Chapter 6 I have sketched a brief account of the theory of Euler products which he started but which really started to unfold with the work of Dirichlet and which, in the course of a long history, finally reached its climactic developments with the work of the number theorists of the late 19th century (class field theory) and the currently very active Langlands program. Parts of this chapter may be regarded as a very brief introduction to the Langlands program.

When writing any book, the question is always this: for whom is the book being written? To a large extent I was guided by a desire to reach the beginning graduate students and the advanced undergraduates, and to communicate to them the marvelous fact that things like class field theory, Borel summation, elliptic curves, and so on, did not come out fully grown from the primeval ocean (of the ancient Hindus for example) but grew out of small beginnings, and many of these go back to Euler. I feel that this historically motivated method of teaching is the best suited to convey the organic structure of mathematics. This of course is not the preferred way of teaching nowadays, where the students learn coherent sheaves and adeles before learning the Euler-Fermat theorem of primes which are sums of two squares. It actually happened in a high-powered course on $D$-modules I was attending a few years back that the first differential equation that was written, a few weeks after some very heavy stuff on $D$-modules, was Euler's, and the lecturer got it wrong because he forgot that the invariant operator on $\mathbb{C}^{*}$ is not $d/dz$ but $z(d/dz)$! I am convinced that the only way to produce young mathematicians who are not imprisoned by the small number of ideas they learn in a conventional graduate education is to emphasise the unity of mathematics from the beginning, and for this, the historical method, called the biogenetic method by Shafaraevitch. is the only possible one.

Thus this book is not a conventional historical essay on Euler of these there are many wonderful examples but rather a discussion of some of the Eulerian themes and how they fit into the modern perspective. More than anyone else, I know what the shortcomings of my attempt are. For example, although I have tried to keep the exposition as elementary as possible, here and there are places where this has been impossible to maintain, and I have had to assume familiarity on the part of the reader of more advanced material. But I have tried to organise everything in such a manner that a beginning graduate student, as well as a mathematician who does not always have a specialised knowledge of the topics treated, will find things understandable as well as enjoyable.

It only remains for me to give thanks to the people who helped me in this effort: to Don Babbitt and Sergei Gel'fand, who kept encouraging me throughout this enterprise; to Marina Eskin, who attended my course on Euler with an enthusiasm that was infectious; to Pierre Deligne, who was extraordinarily generous, as he always is, in sharing with me his insights on many aspects of Euler's work during his visit to UCLA in the spring of 2005; to Anita Colby, science librarian at UCLA, who always gave her help freely, instantly, and with a smile, for her help in bringing many items of Euleriana to my attention; to Richard Tsai for helping me with calculations using MATHEMATICA; to Aaron Pearl for help in photographing pages from Euler's papers; to Boyan Kostadinov, my student, for his help in translating articles by and about Euler from Russian as well as in proofreading; and to my wife, Veda, whose understanding and support have been the great steadying influences in my life.

12.2. Review by: Panjan Roy.
The American Mathematical Monthly 115 (5) (2008), 469-473.

Leonhard Euler was the greatest mathematician of the eighteenth century. His dominance of every aspect of mathematics and its applications was unprecedented and has not been matched. He systematised, organised, and extended diverse results of earlier mathematicians; he solved several outstanding problems which had baffled his most brilliant predecessors and contemporaries; he created many new concepts and areas of research for later generations. His tremendous output fills more than seventy large volumes. Anyone who has taken mathematics courses has seen some work of Euler. It is an impossible task to discuss a large part of Euler's work within its historical perspective in a book of reasonable size. Thus, Varadarajan's aim is much more focused; as he states, "This book is not a conventional historical essay on Euler ... but rather a discussion of some of the Eulerian themes and how they fit into the modern perspective." I think the author has succeeded in fulfilling the task he describes. The book gives the reader an accurate sense of the scope and extent of Euler's work in pure mathematics, a feeling for how he thought, and an outline of selected topics in modern mathematics which have roots in Euler.
...
Varadarajan's book is very well written and motivated. His account of Euler's work is lively and readable. He shows clearly that Euler is an important and pervasive influence in modern mathematics. In this connection, Varadarajan's book provides a very valuable service, in illustrating that the works of Euler are not merely interesting but vibrant, useful and relevant. It may sometimes be overlooked that understanding the historical development of mathematics yields a deeper and clearer appreciation of the mathematics of today. A second year graduate student with a knowledge of number theory should be able to understand almost everything in the book, and an advanced undergraduate could also understand a good amount. I recommend it very highly as a book that mathematicians may learn from and enjoy.

12.3. Review by: Douglas M Jesseph.
Mathematical Reviews MR2219954 (2007h:01009).

This book is a study in the mathematics of Leonhard Euler (1707-1783), with particular attention paid to those aspects of his voluminous work that connect to current mathematical research. As the author notes, he does not intend merely to describe what results Euler achieved and when, but rather to "pay great attention to the historical evolution of ideas and how they mesh with what we know and are interested in today." The result is very much a forward-looking book, which takes up themes from Euler and delineates their connection to current concerns in mathematics. On the whole, it is a satisfying achievement, although some historians of mathematics will find that it is insufficiently concerned with Euler's historical context and the details of his discoveries.

The six chapters include a concise and accurate scientific biography of Euler (Chapter 1); a study of themes in the calculus and number theory (Chapter 2); a detailed look at the Eulerian theory of zeta values (Chapter 3); a treatment of the Euler-Maclaurin sum formula and its applications (Chapter 4); an extensive account of Euler's work on divergent series and its influence on later mathematics (Chapter 5); and a study of the Euler product formula, as well as subsequent research into Euler products and their role in number theory. A useful "portrait gallery" is appended, which gives thumbnail sketches of the careers and contributions of some of the mathematicians whose work is mentioned in the course of the book.

The presentation is clear and to the point, with technical material developed in a way that is generally accessible. Historians of mathematics with a somewhat rusty grasp of nineteenth and twentieth century developments in such subjects as the theory of infinite series and products may find parts of the book rather formidable. Nevertheless, the exposition and development of the key ideas is never inscrutable to someone with a sound undergraduate background in mathematics.

One of the most intriguing chapters of the book is its study of Euler's treatment of divergent series. Lacking our current understanding of such concepts as convergence and uniform convergence, Euler was left to navigate the murky waters of series expansions and infinite sums in ways that offend against current standards of rigor. He had no compunction about taking the assignment of a sum to a divergent series as a matter of more or less arbitrary convention, a willingness which opened up the possibility that different conventions might yield different sums for the same series. Guided by an almost infallible intuition, Euler nevertheless achieved results that withstood scrutiny. In the course of time, the inherent difficulties (indeed, incoherence) in the careless use of divergent series led to conflicting "results" and Abel's famous remark that "Divergent series are in general the work of the devil and it is shameful to base any demonstration whatever on them." Such scruples led to the modern theory of sequences, series, and convergence as developed by Abel and Cauchy, and became the foundation of analysis as we now know it. Varadarajan shows not only how Euler's main results (such as the functional equation of the zeta function) are recoverable in the modern approach to analysis, but that some of his ideas led into altogether new territory. In one case, Euler's work anticipated the theory of Borel summation in the solution of differential equations with asymptotic expansions, and in another his ideas can be traced to current work in summation of divergent Gaussian integrals that appear in quantum mechanics and quantum field theory.

Although not a traditional history of mathematics, Varadarajan's contribution is an interesting addition to the literature on eighteenth century mathematics. By taking some of Euler's most important insights, developing them, and showing their connection to contemporary research, this book offers a profound understanding of Euler's achievements and their role in the development of mathematics as we now know it.

13.1. From the Preface.

The essays in this book collect some of the lectures I have given at various places. Although the lectures ranged over a wide array of topics, they had a single theme which reflected my deep interest in the role of symmetry and supersymmetry in quantum theory. As such they are mathematical as well as personal.

Symmetry and supersymmetry, especially in quantum theory, are expressed in the language of the theory of unitary representations. This is a subject of great intrinsic beauty and enters other parts of mathematics at a very deep level. Two of the greatest figures in its history are Mackey and Harish-Chandra. Their work (to use the words of Weyl) affords shade to large parts of present day mathematics and high energy physics. It is to their memory that this volume is lovingly dedicated.

The essays should perhaps be viewed like a stroll through a garden of ideas: quantum algebras, super geometry, unitary supersymmetries, differential equations, non-archimedean physics, are a few of the topics encountered along the way. My own mathematical education evolved out of interactions with Mackey and Harish-Chandra, and I conclude this volume with brief portraits of their work, embedded in the context of personal reminiscences.

Quite a large number of people have had a hand in shaping my views expressed in these essays. In the essays themselves I have made an effort to mention some of them. But I must acknowledge my deep gratitude here to my longtime friend and collaborator Don Babbitt for the hundreds of discussions on the topics discussed here, spread literally over a lifetime.

I am also indebted to Don Blasius, who, despite many demands, personal as well as professional, on his time, took a deep interest in these essays and made many suggestions for improvement. A first draft of a substantial part of this book was written while my wife Veda and I were vacationing in the Dordogne, living in Don's house in May of 2009. The delights of rural France were a big inspiration for me and so I should thank Don and Peter, as well as Barry and Julie (not to mention Sukie!), for making life so pleasant in Chez Blasius.

Finally, I wish to thank Ann Kostant for the help and encouragement she has given me throughout this enterprise. It was her enthusiastic acceptance of my suggestion, made in the Spring of 2009, that Springer publish a book of essays collecting together some of the lectures I have given at various places, that was the genesis of this book. I really cannot thank her enough.

13.2. Review by: William G Faris.
SIAM Review 54 (1) (2012), 186-190.

This volume is a marvellous collection of essays that marries the technical with the personal. The technical part is an exploration of symmetries that arise in considerations of quantum theory. The personal part consists of reminiscences by the author about his colleagues, in particular the physicist Moshe Flato and the mathematicians George Mackey and Harish-Chandra, This is not a textbook, in that there is no attempt to develop the theory from the beginning; there is not even an index. Instead, the reader should look to these essays for an overview of certain topics: the mathematical structure of quantum theory, symmetry and its generalisation to supersymmetry, non-Archimedean physics, and the structure of differential equations with irregular singularities. Each essay provides a list of readings for those who wish to pursue the subject in greater depth.
...
13.3. Review by: Howard E Brandt.
Mathematical Reviews MR2808564 (2012g:81002).

The distinguished mathematician Veeravalli S Varadarajan shares several of his brilliant, stimulating, and informative mathematical essays, including ones on quantum algebras, supergravity, unitary supersymmetries, differential equations, and non-Archimedean physics. Also, portraits are included of the work and his personal reminiscences of George Mackey and Harish-Chandra, who were both major influences on his own work. Varadarajan also expresses his own views on various aspects of the quantum theory.

The book begins with a Prologue sketching various aspects of quantum mechanics and the author's career in quantum theory and unitary group representation theory, along with his associations with Julian Schwinger, Roger Richardson, Albert Nijenhuis, William Woolf, Mackey, Warren Hirsch, Srinivasa Varadhan, Harish-Chandra, Enrico Beltrametti, Donald Babbitt, Robert Finkelstein, and Sergio Ferrara. Following the Prologue is Chapter 2 on quantum algebras, including Paul Dirac's algebra in quantum mechanics, von Neumann's algebraic treatment of quantum mechanics, Schwinger's insightful measurement algebra and its extension by L Accardi, Hermann Weyl's projective representation and quantisation map, Weyl-Moyal algebra and the Moyal bracket, deformation of associative algebras, and phase space algebras. I was particularly interested in what Varadarajan had to say about Schwinger's measurement algebra, which I have found to be very useful in the analysis of problems in quantum optics and quantum information processing [H E Brandt, (1999)]. The chapter concludes with Varadarajan's reminiscences of Moshe Flato and his treatment, together with his collaborators F Bayen, Christian Frønsdal, Andre Lichnerowicz, and Daniel Sternheimer, of Weyl-Moyal algebra and quantisation in the framework of deformations of Poisson algebra. Varadarajan notes Flato's deep interest in formulating and understanding nonlinear aspects of quantum field theory.

In Chapter 3, Varadarajan offers an essay on the mathematical theory of probability in the quantum world along with discussions of several popular issues in quantum mechanics. He reviews the statistical interpretation of quantum theory and discusses it in terms of von Neumann's propositional logic. Heisenberg's uncertainty principle and wave-particle duality are also considered. The hidden variables question and Bell's inequalities are also addressed. It is argued that there is no way, compatible with experiment and theory, to obtain the results of quantum theory by any reasonable classical hidden variables theory. Also addressed is the Einstein-Podolsky-Rosen question of the possible incompleteness of quantum mechanics. Atomic transition probabilities and Planck's blackbody radiation formula are also considered. Also, the Feynman path integral formulation of quantum mechanics and the Feynman-Kac formula for the quantum propagator are critically examined. An essay on supersymmetry, supermanifolds, and supergravity is the subject of Chapter 4. Varadarajan first briefly reviews the historical evolution of classical geometry, including Riemannian geometry, differential topology, global differential geometry, and algebraic geometry. The concept of a sheaf of rings of numerical functions in place of standard coordinates is introduced to describe the local structure of space. By considering various algebraic operations of the elements of the ring, one obtains various kinds of geometry: algebraic, differential, analytic, holomorphic, etc., and by changing the number system one obtains real, complex, or $p$-adic manifolds. This perspective culminated with Alexandre Grothendieck's theory of schemes in which the local rings are completely arbitrary commutative rings formally attached to regions of space. The spaces themselves appear as the spectra of the rings, namely the domain of elements of the rings. Furthermore the underlying structure of a space is completely determined by the category of sheaves defined on the scheme. The category, where objects represent the open sets and where rings are the possible fundamental rings allowed in the category, replaces the traditional notion of space itself as consisting of points and their topology. The sheaves, viewed as functors from the category, form the topos. The étale toposes then connect topology and differential geometry, the continuous world with the discrete world of number theory. Although sheaves and toposes may be used to describe the commutative world, in quantum mechanics the algebra of physical quantities is in general noncommutative. These quantum algebras may be related to spacetime by means of supersymmetry and may suggest noncommutative aspects of spacetime itself.

Varadarajan traces the development of the concept of supersymmetry and the role of the connection between particle spin and statistics. Supersymmetry arises as transformations that exchange bosonic and fermionic states. Classical geometry is generalised to include noncommuting Grassmann coordinates encoding the fermionic aspects of matter. The automorphisms of the associated supermanifolds are supersymmetries forming super Lie algebras.

There is currently no evidence for supersymmetry in nature, although it is anticipated that it will eventually be seen at energies exceeding those currently available in particle accelerators. Breaking of supersymmetry in the early universe, and extensions of electrodynamics, Yang-Mills theory, and Einstein gravity to include supersymmetry are addressed. Also, a generalisation of Grothendieck's theory of schemes using the functorial method is needed in which the local rings are supercommutative with local structure consisting of a Grassmann algebra over commutative local rings.

Varadarajan gives a brief historical perspective on the introduction of supersymmetry by physicists and mathematicians. He notes that in early papers Schwinger introduced Grassmann algebras in fermionic quantum field theory. Physicists contributing to the development of supersymmetry included Abdus Salam, J Strathdee, Ferrara, Julius Wess, Bruno Zumino, Yuri Golfand, Evgeny Likhtman, D V Volkov, and V P Akulov. Mathematicians included P Deligne, J Morgan, B Kostant, D Leites, Yu Manin, and Varadarajan himself.

Varadarajan proceeds to exposit a precise mathematical formulation of the concept of a supermanifold with local commutative and Grassmann coordinates by generalising the definition of a classical manifold with the idea of a ringed space and using the methods of Grothendieck. He describes the category of ringed spaces and their morphisms. For a supermanifold, the commutative local rings are replaced by supercommutative local rings having a local splitting as an algebra over commuting and anticommuting local coordinates. Real, complex, or $p$-adic analytic supermanifolds can be formulated by replacing the classical manifold by a real, complex, or $p$-adic analytic one leading to the concept of a superscheme. The category of super vector spaces and superalgebras is delineated. Super Lie algebras are also examined. A super Lie algebra is a Lie algebra in the tensor category of super vector spaces. For super linear algebra and the theory of modules over supercommutative rings, a generalisation of the determinant, namely the Berezinian, is needed. The author proceeds to consider the functor of points and morphisms in supermanifolds. He describes the differential and integral calculus and change of variables in supermanifolds. Also, the global section functor is given as a functor from the category of smooth supermanifolds to the category of superalgebras. A heuristic conception of supermanifolds is offered as a classical smooth manifold surrounded by a Grassmannian cloud. Super Lie groups acting on supermanifolds are also considered. Super Harish-Chandra pairs are also described as the classical Lie group underlying the super Lie group together with the corresponding super Lie algebra. This yields a covariant functor which is an equivalence of categories from the category of super Lie groups to the category of super Harish-Chandra pairs. The action of super Lie groups on supermanifolds can be described in terms of Harish-Chandra pairs.

Chapter 5 is an essay on unitary representations of super Lie groups in which the author draws heavily on his own work. Considered are the super imprimitivity theorem for homogeneous spaces and unitary irreducible representations of supermanifold products. Germane to superfield theory in physics, Varadarajan also addresses super Poincaré Lie algebras and Lie groups, unitary representations of super Poincaré groups, and superparticles and their multiplet structure. It is noted that the threshold for supersymmetry breaking in the early universe remains unknown, and the predicted superpartners of elementary particles remain to date unseen.

Non-Archimedean physics is the subject of Chapter 6. Following the origins and description of the Planck scale, Varadarajan discusses the speculative proposal of Igor Volovich in 1987 that the geometry of spacetime may be non-Archimedean in sub-Planckian domains. Beltrametti had suggested earlier that the geometry of spacetime might be based on non-Archimedean or finite fields. It is currently generally recognized that the classical spacetime structure breaks down at the Planck scale. Non-Archimedean geometries are defined over the field of $p$-adic numbers, where $p$ is a rational prime exceeding two. Manin later proposed that an adelic geometry be considered as the basis for spacetime. Elementary particles are supposedly defined by irreducible projective representations of the $p$-adic analogs of the real Poincaré and Galilean groups. One then has to define the spacetime symmetry groups as $p$-adic Lie groups and consider their projective representations. Varadarajan speculates with little justification that Planck-scale fluctuations of spacetime may change the ground field from real numbers to $p$-adic numbers. He proceeds to consider non-Archimedean symmetry groups and their multipliers. Mackey's formula for multipliers of semidirect products is introduced. Elementary particles defined over non-Archimedean fields are also discussed, and it is argued that particles defined over $p$-adic numbers have a richer structure than those defined over the real numbers. It is pointed out that the determination of the multiplier of the little groups remains incomplete. Varadarajan also describes how Galilean symmetry and Galilean particles can be treated for $p$-adic numbers. He also considers possible extensions of conformal symmetry and conformal spacetime for $p$-adic numbers, and notes that further development is needed. He considers nonrelativistic quantum mechanics with kinematical and dynamical objects having $p$-adic manifolds as sources, and a generalisation of the Feynman-Kac path integral is given. Varadarajan mentions that the extension of supersymmetry to the $p$-adic setting remains an open problem, and he suggests that representations in $p$-adic Banach space may be needed. Finally, the author considers the possibility of non-Archimedean analogs of quantum fields and also particle scattering theory.

Chapter 7 consists of an essay on differential equations with irregular singularities for which formal solutions do not always converge. Drawing largely on his own work together with Babbitt, Varadarajan relates irregular linear differential equations to topics in commutative algebra, linear algebraic groups, and algebraic geometry. He focuses on questions regarding irregular singular solutions which have a group theoretic origin.

The book concludes with an essay on the contributions of Mackey, Harish-Chandra, and the author to group representation theory. Varadarajan reminisces on his associations with Mackey and Harish-Chandra. He argues that Mackey's program was largely to determine the unitary irreducible representations of all locally compact second countable groups and to decompose important representations into irreducible constituents. Mackey uncovered surprising aspects of the representation theory of these groups and their relationship to sophisticated aspects of measure theory. He also developed a beautiful theory of central extensions of these groups. He was the first to advocate that ergodic action merits major investigation. Mackey also significantly impacted work on semisimple groups. Harish-Chandra's contributions to harmonic analysis on semisimple groups, and the discrete series are next considered. He recognised that because the groups are noncompact, one should implement differential rather than integral operators. The construction of the discrete series is among his greatest achievements, as this is the key to the construction of the irreducible representations of semisimple groups that occur in the regular representation and leads to an explicit description of the Plancherel measure.

14.1. Form the Publisher.

The papers reproduced in these two volumes include not only the articles published since 1998 but also some important papers published earlier which were not included in the original Selected Papers for reasons of space and other limitations.

The current volumes contain the papers on fundamental questions of individual and families of meromorphic differential equations that are treated by a new group theoretic and functional approach; papers on representation theory of Lie groups; papers on foundations of physics, supersymmetry, and $p$-adic aspects of quantum physical theories; papers on analysis, especially oscillatory integrals, on semi-simple lie groups, their conjugacy classes, and their flag manifolds; and finally, several review articles, both personal and mathematical, on a number of the above topics.

The volumes are divided into sections by topics and in each section the editors have included a brief summary of the contents of the section as well as putting the papers in the historical context and, in the process, raising interesting new problems.

14.2. Review by: Anargyros Fellouris.
Mathematical Reviews MR3157467.

This book is a collection of nine papers of V S Varadarajan. In addition to the papers themselves, this volume contains a complete list of his publications between 1958 and 2012 consisting of 14 books, 3 book reviews and 121 papers. The papers included in this volume cover two main parts: differential equations and representation theory. In the first part the emphasis is on the local reduction theory and moduli of meromorphic differential equations. The papers presented are mainly in collaboration with Donald G Babbitt. The second group of papers is devoted to the field of representation theory. It starts with an article of Ramesh Gangolli on Varadarajan and representation. It contains comments on the papers presented in this volume and on the personality of Varadarajan. Two of the papers presented are devoted to the great mathematicians Harish-Chandra and George Mackey, two of the most influential figures in the history of representation theory. These reports are a retrospective of their work and its influence. The main effort is to sketch briefly the outlines of their work and their main ideas for the benefit of a wide mathematical audience as well as younger mathematicians.

15.1. Form the Publisher.

See 14.1 above.

15.2. Review by: Daniel Beltita.
Mathematical Reviews MR3157468.

The volume under review contains about thirty articles by V S Varadarajan, divided according to five main themes: 1. Foundations of quantum theory; 2. Supersymmetry; 3. $p$-adic physics; 4. Analysis and probability; 5. Reviews and reflections. The first three of these groups of papers are prefaced by very useful introductory articles by the editors of this volume. Thus, E G Beltrametti and G Cassinelli write about "Varadarajan and the Department of Physics of the University of Genoa", R Fioresi about "V S Varadarajan and supersymmetry", and A N Kochubei about "V S Varadarajan and $p$-adic mathematical physics". This volume includes some of the author's seminal papers in the aforementioned areas, as for instance the paper by C Carmeli et al. (2006) or the paper by J J Duistermaat, J A C Kolk and Varadarajan (1983). Besides the mathematical papers, a quite instructive perspective is offered by the author's reflections in the articles included in the last part of this volume, on the mathematicians H A Dye, R F Arens, Harish-Chandra, N V Pedersen, M Flato, A Weil, D W Strook, S S Varadhan, and G W Mackey.