1.1. From the Preface.

The following pages give a detailed account of the course given by Professor Henri Cartan at Harvard University during the spring of 1948. The editors wish to express their most sincere thanks to Professor Henri Cartan for the permission to publish these notes, and in particular for his confidence in allowing them to be prepared in his absence. We hope that we have lived up to this trust, and have not committed too many errors in the process.

Since the time for these lectures was limited, many details have been supplied by the editors; many thanks are also due to Messrs Paul Olum, Maxwell Rosenlicht, and Lawrence Markus for working out some of the missing proofs. Finally, we wish to express our gratitude to Professor Lars V Ahlfors and the Department of Mathematics at Harvard for their cooperation in this project.

1.2. Review by: S Chern.
Mathematical Reviews MR0030753 (11,46e).

The work consists of notes based on a course given at Harvard University in 1948. The essential feature is a comprehensive homology (or cohomology) theory of locally compact topological spaces, with the notion of grating playing the central role. Although a notion due to Alexander and carrying the same name is included as a particular case, the present treatment is characteristically axiomatic. After four chapters dealing with the basic concepts of algebraic topology and exterior differential forms, the author proceeds to consider graded rings $A$ with a differential operator $d$ such that $d^{2} = 0$. With respect to $d$ the derived ring is defined. This notion will be given a topological significance when the supports of elements in $A$ are defined. The latter are functions in $A$, with values which are closed subsets of a locally compact topological space, satisfying certain conditions. A grating is a graded ring with differential operator and supports. It is shown that this theory includes the de Rham theory of differential forms on a differentiable manifold, the Alexander grating theory, the Čech and singular homology theories. The coefficient ring $\mathfrak{A}$ comes into play by considering it as a grating with every element having degree zero, with vanishing differential operator, and with supports defined to be either the whole space or empty. Call $A$ an $\mathfrak{A}$-grating if $\mathfrak{A}$ can be identified as a subgrating of $A$. If, moreover, $\mathfrak{A}$ is isomorphic to the derived ring $H(A/A_{M})$ at every point $M$ of $E$, $A$ is called $\mathfrak{A}$-simple. The culminating result of this work is an isomorphism theorem which characterises the cohomology ring of a compact space and may be stated as follows.
Let $E$ be a compact space, $A,B$,$\mathfrak{A}$-grating and $\mathfrak{B}$-grating, respectively, which are fine and are $\mathfrak{A}$-simple and $\mathfrak{B}$-simple, respectively. If there is an isomorphism $f$ of f onto $\mathfrak{B}$, the induced homomorphism $\hat{f}$ of $H(A)$ into $H(B)$ is an isomorphism onto.

2.1. From the Preface.

During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to present a unified account of these developments and to lay the foundations of a full-fledged theory.

The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments. We present herein a single cohomology (and also a homology) theory which embodies all three; each is obtained from it by a suitable specialisation.

This unification possesses all the usual advantages. One proof replaces three. In addition an interplay takes place among the three specialisations; each enriches the other two.

The unified theory also enjoys a broader sweep. It applies to situations not covered by the specialisations. An important example is Hilbert's theorem concerning chains of syzygies in a polynomial ring of n variables. We obtain his result (and various analogous new theorems) as a theorem of homology theory.

The initial impetus which, in part, led us to these investigations was provided by a problem of topology. Nearly thirty years ago, Künneth studied the relations of the homology groups of a product space to those of the two factors. He obtained results in the form of numerical relations among the Betti numbers and torsion coefficients. The problem was to strengthen these results by stating them in a group-invariant form. The first step is to convert this problem into a purely algebraic one concerning the homology groups of the tensor product of two (algebraic) complexes. The solution we shall give involves not only the tensor product of the homology groups of the two complexes, but also a second product called their torsion product. The torsion product is a new operation derived from the tensor product. The point of departure was the discovery that the process of deriving the torsion product from the tensor product could be generalised so as to apply to a wide class of functors. In particular, the process could be iterated and thus a sequence of functors could be obtained from a single functor. It was then observed that the resulting sequence possessed the formal properties usually encountered in homology theory.

2.2. Review by: G Hochschild.
Mathematical Reviews MR0077480 (b).

The title "Homological Algebra" is intended to designate a part of pure algebra which is the result of making algebraic homology theory independent of its original habitat in topology and building it up to a general theory of modules over associative rings. The particular formal aspect of this theory stemming from algebraic topology is that of a preoccupation with endomorphisms of square 0 in graded modules. The conceptual flavour of homological algebra derives less specifically from topology than from the general 'naturalistic' trend of mathematics as a whole to supplement the study of the anatomy of any mathematical entity with an analysis of its behaviour under the maps belonging to the larger mathematical system with which it is associated. In particular, homological algebra is concerned not so much with the intrinsic structure of modules but primarily with the pattern of compositions of homomorphisms between modules and their interplay with the various constructions by which new modules may be obtained from given ones.

In the recent requisite mathematical terminology, this means that the principal objects of study in homological algebra are functors from categories of modules to other categories of modules.
...
The appearance of this book must mean that the experimental phase of homological algebra is now surpassed. The diverse original homological constructions in various algebraic systems which were frequently of an ad hoc and artificial nature have been absorbed in a general theory whose significance goes far beyond its sources. The basic principles of homological algebra, and in particular the full functorial control over the manipulation of tensor products and modules of operator homomorphisms, will undoubtedly become standard algebraic technique already on the elementary level. It is probably with such expectations that the authors have put so much missionary zeal into the systematisation of their approach and the cataloguing of the basic results. A probably unavoidable effect of this is that the book cannot be consulted by the uninitiated in a local fashion. The reader is definitely forced to go through it starting at the beginning. Each chapter (with the exception of the last) is followed by a collection of exercises which are designed not so much to strengthen the reader by easy gymnastics (for they are generally not particularly easy) as to point out various ramifications and applications of the general theory.

2.3. Review by: I M James.
The Mathematical Gazette 41 (338) (1957), 310-311.

Since the war there have been massive developments in algebraic topology, and certain branches have attained a high degree of independence from the mother subject. The most remarkable of these has now received a monumental exposition in the hands of Henri Cartan and Samuel Eilenberg. This long-awaited book consists of an entirely new mathematical theory, about which little has previously been published. To quote from the Preface:

During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to present a unified account of these developments and to lay the foundations of a fully-fledged theory.

The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments. We present herein a single cohomology (and also homology) theory which embodies all three; each is obtained from it by a suitable specialisation.

This unification possesses all the usual advantages. One proof replaces three. In addition an interplay takes place among the three specialisations; each enriches the other two.

The unified theory also enjoys a broader sweep. It applies to situations not covered by the specialisations. An important example is Hilbert's theorem concerning chains of syzygies in a polynomial ring of n variables. We obtain his result (and various analogous new theorems) as a theorem of homology theory.

The initial impetus which, in part, led us to these investigations was provided by a problem of topology. Nearly thirty years ago, Künneth studied the relations of the homology groups of a product space to those of the two factors. He obtained results in the form of numerical relations among the Betti numbers and torsion coefficients. The problem was to strengthen these results by stating them in group-invariant form. The first step is to convert this problem into a purely algebraic one concerning the homology groups of the tensor product of two (algebraic) complexes. The solution we shall give involves not only the tensor product of the homology groups of the two complexes, but also a second product called their torsion product. The torsion product is a new operation derived from the tensor product. The point of departure was the discovery that the process of deriving the torsion product from the tensor product could be generalised so as to apply to a wide class of functors. In particular, the process could be iterated and thus a sequence of functors could be obtained from a single functor. It was then observed that the resulting sequence possessed the formal properties usually encountered in homology theory.

The work is self-contained, and requires less an extensive knowledge of algebra than an ability to keep in mind the quantities of new functors which the authors keep introducing. There is an absence of external motivation throughout, and the topological background is hardly mentioned at all. The work is to be appreciated first of all as a piece of pure, creative algebra, but towards the end of the book there are chapters on the applications. These refer to finite groups, with class-field theory especially in mind, Lie algebras, and the classification of extensions for groups, algebras, and modules. There is a useful exposition of spectral sequences, which is somewhat independent of the rest of the book. In an appendix, which might very well be read before the book itself is begun, David Buchsbaum places the whole work in a more general setting with his theory of exact categories. There is a good index, and an invaluable list of the hundred-odd symbols which Eilenberg and Cartan use with a special meaning.

2.4. Review by: Saunders MacLane.
Bull. A mer. Math. Soc.62 (6) (1956), 615-624.

At last this vigorous and influential book is at hand. It took nearly three years from completed manuscript to bound book; Princeton is penalised 15 yards for holding.

Homological algebra deals both with the homology of algebraic systems and with the algebraic aspects of homology theory. The first topic includes the homology and cohomology theories of groups, of associative algebras, and of Lie algebras. The second topic includes the care and feeding of exact sequences and spectral sequences, as well as the manipulation of functors of chain complexes. ...
...
In spite of the delay in its publication, widespread acquaintance with the manuscript and with the ideas of this book has already played an important role in the development of this lively subject. ...
...
The authors' approach in this book can best be described in philosophical terms and as monistic: everything is unified. Consider for instance the homology of groups; in view of its application to class field theory and to topology this topic is central in homological algebra. In this book the homology of groups appears as a special case of the homology of monoids (monoid = associative multiplicative system with identity), which in turn is a special case of the homology of supplemented algebras, again a case of the homology of augmented algebras, which is an instance of a torsion product, which at your choice is an instance of a derived functor or an iterated satellite functor.

Historically, each monistic doctrine is resolved by a subsequent pluralism. So it was here. When the authors started to write, it was true that all known cases of homology of algebraic systems (groups, algebras, and Lie algebras) could be neatly subsumed under the resolution, Tor, and Ext pattern. When they finished writing this was no longer so-and this because of the authors' own separate efforts elsewhere! The Eilenberg-MacLane homology of abelian groups (Trans. Amer. Math. Soc. vol. 71 (1951) pp. 294-330) has not yet been expressed by torsion products. The Eilenberg-Mac Lane bar construction (Annals of Math. vol. 58 (1953) pp. 55-106) is a standard construction more general than those produced by standard resolutions. Cartan's beautiful and powerful theory of constructions (Séminaire École Normale Supérieure, 1954/1955) is an extension of the idea of a projective resolution beyond the terms of this book. Still more recently, the as yet unpublished homology theories of Dixmier for Lie rings and of MacLane for rings are other examples of homology of algebraic systems not (at least as yet) obtainable by resolutions.

Perhaps Mathematics now moves so fast and in part because of vigorous unifying contributions such as that of this book-that no unification of Mathematics can be up to date. The reviewer might also add his strictly personal opinion that the authors have not kept sufficiently in mind the distinction between a research paper and a book: a good research paper presents a promising new idea when it is hot - and when nobody knows for sure that it will turn out to be really useful; a good research book presents ideas (still warm) after their utility has been established in the hands of several workers. This book contains too large a proportion of shiny new ideas which have nothing to recommend them but their heat and promise: satellites (these appear in Chapter III and then gradually disappear in later chapters), derived functors of anything but Hom and (the reviewer watched in vain for other examples), semi-hereditary rings, functors derived simultaneously in several variables, supplemented algebras, and the homology of monoids. The same remark applies to spectral sequences. These sequences have proved their worth in topology but have not yet reached decisive results in the homology of algebraic systems: the result is that the uninitiated reader can hardly hope to understand what spectral sequences are all about by reading the three chapters devoted to them in this book. The reviewer is not claiming that spectral sequences and these other notions will not later have significant algebraic uses: some of them will, but until that time comes their presence clutters up the book.
...
The authors' treatment of the literature is off-hand. ...

2.5. Review of the 1999 reprint by: Brian Denton.
The Mathematical Gazette 84 (500) (2000), 359-360.

When asked to review this book for the Gazette, nostalgia ran wild! I immediately took my hardback copy from my shelf, dusted it down and considered when I first bought the book. Page for page this new edition in paperback is identical to the original hardback first published in 1956 (the edition I admit I have, but I was a mere youngster then!). It is a classic text and hence it now appears in the Princeton Landmarks series.

Henri Cartan was Professor of Mathematics at the University of Paris and is a Fellow of the Royal Society. He, like many other pure mathematicians in this country, is also a member of the American Mathematical Society. He was made an Honorary Member of the London Mathematical Society in 1959. Samuel Eilenberg was Professor of Mathematics at Columbia University and died in 1998. Both of these eminent mathematicians were founding members of the Bourbaki and both received the Wolf Prize in Mathematics.

So what can I say about this 'masterpiece'? The book is for pure mathematicians working in the area of algebraic topology. (It is not one that the majority of Gazette readers will be interested in. I seem to say this in many of the reviews I write.) Its starting point is rings and modules. From the 390 pages the first 13 pages of mathematics (Chapter 1), cover projective and injective modules, semi-simple, hereditary and semi-hereditary rings as well as Noetherian rings. These are the basic objects for the subject. There are seventeen chapters, the remaining sixteen having titles: Additive Functors; Satellites; Homology; Derived Functors; Derived functors on and Hom; Integral Domains; Augmented Rings; Associative Algebras; Supplemented Algebras; Products; Finite Groups; Lie Algebras; Extensions; Spectral Sequences; Applications of Spectral Sequences and, finally, Hyperhomology. There are exercises at the end of chapters and there is an appendix called Exact Categories written by David Buchsbaum. After the appendix, a useful list of symbols is given with a page reference as to when the symbol is first introduced and explained.

Originally this book was number 18 in the Princeton Mathematical Series. If the Princeton University Press reproduces all of that original series it would be doing the mathematical fraternity a big service, as present day mathematicians could then collect them at the reasonable price at which they are being published.

Thank you to the Book Reviews Editor for stirring my original pure mathematics days of long ago!!

3.1. From the Publisher.

This is a work that has stood the test of time. The author, Henri Cartan, a professor at the University of Paris, left a significant mark on 20th-century mathematics. This book is based on one of his undergraduate courses. Primarily devoted to the study of analytic functions of a complex variable (holomorphic functions, Cauchy's integral formula, Morera's theorem, Laurent and power series expansions, the residue theorem, sequences of holomorphic and meromorphic functions, series, infinite products, holomorphic transformations, conformal representation, etc.), it nevertheless addresses the case of one or more real or complex variables, allowing him to discuss holomorphic differential systems in the final chapter. Classical and significant examples are fully explored (the study of the Weierstrass function and the calculation of integrals using the residue theorem through several concrete cases), and exercises complement each chapter. This undergraduate course may also be of interest to those preparing for the agrégation examination.

3.2. Review by: R C Buck.
Mathematical Reviews MR0147623 (26 #5138).

This is an excellent compact treatment of the basic ideas and techniques of elementary complex analysis. Economy is achieved by appropriate use of algebraic and topological concepts. The approach is that of Weierstrass, analytic functions being those that are locally power series. Chapter I treats formal power series over a field $K$, specialises K to $\mathbb{R}$ or $\mathbb{C}$ to discuss convergence, and discusses operations with power series, and special functions; analyticity is defined, and the nature of the set of zeros and poles for a meromorphic function is discussed. Chapter II reviews the integration of differential forms, defines f to be holomorphic if f′ exists, proves the Cauchy theorem [f(z)dz is closed], and the usual consequences of it, completed in Chapter III with the theory of residues. Chapter IV discusses analytic functions of several variables, harmonic functions, holomorphic functions of several variables. Hartog's theorem is stated. Chapter V is an excellent modern summary of the classical material dealing with convergence of series and sequences of analytic functions, presented as an examination of the linear space of functions holomorphic in a region, under the compact-open topology. Chapter VI deals with mappings, proves the mapping theorem, and gives a brief introduction to the theory of analytic spaces and Riemann surfaces. Chapter VII deals with continuity and continuation problems for differential equations. The book has a good collection of exercises.

4.1. From the Preface.

The present volume contains the substance, with some additions, of a course of lectures given at the Faculty of Science in Paris for the requirements of the licence d'enseignement during the academic sessions 1957-1958, 1958-1959 and 1959-1960. It is basically concerned with the theory of analytic functions of a complex variable. The case of analytic functions of several real or complex variables is, however, touched on in chapter IV if only to give an insight into the harmonic functions of two real variables as analytic functions and to permit the treatment in chapter VII of the existence theorem for the solutions of differential systems in cases where the data is analytic.

The subject matter of this book covers that part of the "Mathematics II" certificate syllabus given to analytic functions. This same subject matter was already included in the "Differential and integral calculus " certificate of the old licence.

As the syllabuses of certificates for the licence are not fixed in detail, the teacher usually enjoys a considerable degree of freedom in choosing the subject matter of his course. This freedom is mainly limited by tradition and, in the case of analytic functions of a complex variable, the tradition in France is fairly well established. It will therefore perhaps be useful to indicate here to what extent I have departed from this tradition. In the first place I decided to begin by offering not Cauchy's point of view (differentiable functions and Cauchy's integral) but the Weierstrass point of view, i.e. the theory of convergent power series (chapter I). This is itself preceded by a brief account of formal operations on power series, i.e. what is called nowadays the theory of formal series. I have also made something of an innovation by devoting two paragraphs of chapter vi to a systematic though very elementary exposition of the theory of abstract complex manifolds of one complex dimension. What is referred to here as a complex manifold is simply what used to be called a Riemann surface and is often still given that name; for our part, we decided to keep the term Riemann surface for the double datum of a complex manifold and a holomorphic mapping of this manifold into the complex plane (or, more generally, into another complex manifold). In this way a distinction is made between the two ideas with a clarity unattainable with orthodox terminology. With a subject as well established as the theory of analytic functions of a complex variable, which has been in the past the subject of so many treatises and still is in all countries, there could be no question of laying claim to originality. If the present treatise differs in any way from its forerunners in France, it does so perhaps because it conforms to a recent practice which is becoming increasingly prevalent: a mathematical text must contain precise statements of propositions or theorems statements which are adequate in themselves and to which reference can be made at all times. With a very few exceptions which are clearly indicated, complete proofs are given of all the statements in the text. The somewhat ticklish problems of plane topology in relation to Cauchy's integral and the discussion of many-valued functions are approached quite openly in chapter II. Here again it was thought that a few precise statements were preferable to vague intuitions and hazy ideas. On these problems of plane topology, I drew my inspiration from the excellent book by L Ahlfors (Complex Analysis), without however conforming completely with the points of view he develops. The basic concepts of general Topology are assumed to be familiar to the reader and are employed frequently in the present work; in fact this course is addressed to students of 'Mathematics II' who are expected to have already studied the 'Mathematics I' syllabus.

I express my hearty thanks to Monsieur Reiji Takahashi, who are from experience gained in directing the practical work of students, has consented to supplement the various chapters of this book with exercises and problems. It is hoped that the reader will thus be in a position to make sure that he has understood and assimilated the theoretical ideas set out in the text.

4.2. Review by: Gerald R MacLane.
Mathematical Reviews MR0154968 (27 #4911).

This is a translation of Théorie élémentaire des fonctions analytiques d'une ou plusieurs variable complexes. The translation is good in general, but there are a few surprises. For instance ... Aside from these and a few obvious misprints, the translation is accurate and adequate.

4.3. Review by: R P Jerrard.
Pi Mu Epsilon Journal 4 (1) (1964), 23.

The licence d'enseignement is a degree roughly comparable to the B.A., but requires essentially only the study of mathematics. This volume is based upon lectures given by the author at the University of Paris in the theory of analytic functions of a complex variable for the requirements of this degree, and is at the advanced undergraduate or beginning graduate level for American Students. The basic concepts of general topology are assumed to be familiar to the reader.

The exposition is clear and concise. All theorems are given exact statements and (with few exceptions) complete proofs. There is very little heuristic argument or general discussion of ideas. The first three chapters are on power series and integral theorems and their applications. The remaining four chapters are on analytic functions of several variables, sequences of holomorphic functions, holomorphic transformations, and holomorphic systems of differential equations. This classical material is given a modern flavour. There are, for example, sections on the topology of the vector space of continuous (complex-valued) functions in an open set (Chapter V), and on the integration of differential forms on a complex manifold (Chapter VI).

This is an excellent book which gives a clear and lucid presentation of these ideas.

4.4. Review by: R P Boas Jr.
Science, New Series 144 (3619) (1964), 697.

This is a very attractive book for mathematicians, especially for those who are sympathetic with Bourbaki and familiar with his terminology. It presents the essentials of its topic elegantly, accurately, and concisely, using modern ideas and methods very effectively. It is not, at least at present, for the casual reader who wants to look up a reference or refresh his memory. The subject matter is standard - how could it be otherwise! - but the words are unfamiliar, and no concessions are made to the uninitiated. In many cases the new terminology is really justified, since it lets a result appear as simply a special case of a familiar theorem in, say, general topology or algebra; in other cases there seems to be no obvious reason for the change. The subject is treated almost entirely as an end in itself; there are no indications that it can be used outside of pure mathematics and hardly any that it can be used anywhere else in pure mathematics.

Whereas most introductory texts be-gin with differentiable (holomorphic) functions, Cartan approaches the subject via power series: first he does everything possible with formal power series, then fixes attention on the convergent ones. Integrals come next. The index of a closed path with respect to a point is first defined by integration; this allows the author to handle the topological problems easily before he comes to Cauchy's theorem. Next come analytic functions in more than one variable (unusual material for an introductory text) done very briefly, the author's intention being to illuminate the theory of harmonic functions and to prepare the necessary material for discussing analytic differential equations. After this we meet sequences of analytic functions and conformal mapping. The proof of the "Riemann mapping theorem" (a phrase not used in the book) is remarkably concise and transparent. Riemann surfaces are introduced via one-dimensional abstract complex manifolds (these are Riemann surfaces in the classical sense); Cartan prefers to save the name "Riemann surface" for a complex manifold endowed with a holomorphic mapping. Next we have an ac-curate discussion of analytic continuation, which would hardly be possible in a more conventional text. The book ends with proofs of the existence and fundamental properties of analytic solutions of analytic differential equations.

4.5. Review by: F E J Linton.
The American Mathematical Monthly 72 (2) (1965), 220-221.

This is a translation, by John Standring and H B Shutrick, of Cartan's monograph Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Collection Enseignement des Sciences, Hermann, Paris, 1960. Comments are therefore in order as well on the content as on the translation.

Analyticity is viewed, with Weierstrass, as the property of having, at each point, a convergent power series expansion. This enables Cartan to treat real - as well as complex - analytic functions in the same breath. Holomorphy is taken to be the property (of a function of complex variables) of having a complex derivative with respect to each variable. The equivalence of these two notions is proved after the presentation of useful algebraic results on formal power series and a (somewhat jazzed-up) Ahlfors-like version of winding numbers.

The usual elementary results on analytic functions (Cauchy integral theorem, Laurent expansion, residues, maximum modulus, d'Alembert's (fundamental) theorem (of algebra), relations with harmonic functions, Dirichlet problem, etc.) are followed by chapters on: the topology of the spaces of functions continuous, holomorphic, or meromorphic, respectively, on a given domain; holomorphic transformations, conformal mapping, complex manifolds, and Riemann surfaces; and analytic systems of differential equations.

Each of the seven chapters is followed by numerous exercises prepared by M Reiji Takahashi. The clear and economical presentation of the text is neatly complemented by this opportunity to try one's own hand.

As for the translation itself, one has small bones to pick. Two of the eighteen errata listed in the original edition recur here. Among noteworthy examples of mistranslation or poor mathematical English we mention only the beginning of the last paragraph of the preface, the statement of Liouville's theorem, the use of "revision" for "review", and the introductory remarks in the section on Weierstrass's $\wp$-functions. Fortunately, the mathematical thought is nowhere more than briefly obscured, if at all.

5.1. From the Preface.

M Henri Cartan, professor at the Faculty of Sciences in Paris, was born in 1904 in Nancy. His mathematical work focuses on analytic functions of several variables, algebra, and topology. M Cartan is notably the author of the Elementary Theory of Analytic Functions of One or More Complex Variables, published in 1961.

M Cartan's manuscript was received in September 1966. Mrs C Buttin and Messrs F Rideau and J L Verley were responsible for writing the exercises, which were completed in April 1967. The book was finished printing on 13 October 1967.

5.2. From the Preface by Joseph Kouneiher to a later reprint.

Functional analysis is the branch of mathematics, and more specifically of analysis, that studies function spaces. Its historical roots lie in the study of transformations such as the Fourier transform and in the study of differential equations.

The term "functional" originated in the context of the calculus of variations, to designate functions whose arguments are other functions. Its use was generalised to new fields by the Italian mathematician and physicist Vito Volterra. The Polish mathematician Stefan Banach is often considered the founder of modern functional analysis.

The aim of this book is twofold: to introduce the methods underlying differential calculus and differential equations in Banach spaces, and to familiarise the reader with the concepts of differential forms, elements of the calculus of variations, and applications of the moving frame method to curves and surfaces. It begins with a review of the prerequisites necessary to approach the concepts presented in the following chapters: Banach spaces, differentiable maps, finite increments, implicit functions, rank theorems, higher-order differentials, differentiable convex functions, ruled functions, Taylor series, relative extrema, differential equations, differential forms, and moving frames.

Banach spaces were created by Banach around 1930 to solve systems of equations with infinitely many variables. It is a fairly natural framework for solving a problem where the number of unknowns is not specified, such as the vector space of functions.

Banach spaces are defined as normed complete vector spaces. In other words, a Banach space is a vector space V over the real or complex numbers, equipped with a ||.|| norm such that every Cauchy sequence (with respect to the metric $d(x, y) = ||x - y||)$ in $V$ has a limit in $V$. Since the norm induces a topology on the vector space, a Banach space is an example of a topological vector space.

This book also deals with differential forms and uses them to investigate some local and global aspects of the differential geometry of surfaces. Geometric calculus and differential forms have a common origin in Grassmann algebra; but having had different historical developments, it took mathematicians some time to recognise that they both belong to the same mathematical system.

The importance of Cartan's theory of exterior differential forms lies in the fact that differential forms are perfectly defined objects in the context of functional substitution and pull-back (SF-PB) relating to differentiable $C^{1}$ maps from an initial manifold of variables of dimension $M$ to a final state or a manifold of dimension $N$. It is not necessary for the map to have an inverse (or an invertible Jacobian since the dimensions of the domain and codomain are different).

This contrasts sharply with the theory of tensor analysis where constraints must be imposed since maps and their inverses, as well as differential maps and their inverses, are required.

These maps are called diffeomorphisms and are generally used by physicists under the name of coordinate transformations. A tensor that vanishes on the final state is also zero on the initial state, but for tensors these states must be linked by diffeomorphisms. Thus, differential forms transcend tensor calculus, while including it as a special case.

Note that differential forms unify and simplify multivariable calculus. Students interested in this subject will gain a better conceptual understanding from this book than that provided by conventional approaches.

Finally, in the last chapter, the author develops É Cartan's method of moving frames to study the local differential geometry of surfaces embedded in $\mathbb{R}^{3}$ as well as the intrinsic geometry of surfaces.

In mathematics, a moving frame is an extension of the notion of an ordered basis of a vector space, which is often used to study the extrinsic differential geometry of continuous varieties embedded in a homogeneous space. Moving frames were first introduced by Gaston Darboux in the 19th century, for his studies of the Frenet-Serret frame of a curve embedded in a Euclidean space. Subsequently, they were brought to maturity by Élie Cartan and others to study subvarieties of more general homogeneous spaces¹ (such as projective space).

Élie Cartan's method of moving frames is based on the idea of ​​considering a moving frame adapted to the particular problem being studied. For example, given a curve in space, the first three vectors derived from the curve can generally provide a frame at one of its points. More generally, moving frames can be seen as sections of the principal bundle over the open sets U. Cartan's general method exploits this abstraction by using the notion of Cartan connection.

The most common case of a moving frame is that of a bundle of tangent frames on a manifold. In this case, the moving tangent frame on a manifold $M$ is composed of a collection of vector fields $X_{1}, X_{2}, ..., X_{n}$ forming a tangent base at each point of the open set $U . M$.

It is always possible to define a moving frame locally, i.e., in the neighbourhood $U$ of any point $p$ in $M$; however, the existence of a global moving frame on $M$ requires topological conditions. For example, when $M$ is a circle or, more generally, a torus, this type of frame can be defined; however, this is no longer possible when $M$ is a 2-sphere. A manifold that has a global moving frame is called parallelisable.

The book's content covers both pure and applied branches of mathematics. It develops progressively according to an approach that should be particularly useful to students preparing for conour examinations, or taking differential calculus modules at the master's level, as well as to teachers. It is also of great use to engineering students and theoretical physicists. The text is illustrated with examples, and each part includes a series of exercises of varying difficulty.

This book can be considered a reference and a brilliant mathematical exercise.

5.3. Review by: The Editors.
Mathematical Review MR0223194 (36 #6243).

This is a modern text on calculus and differential equations, and beautifully presents these subjects to the reader in a geometrical coordinate-free manner. The setting is in Banach spaces, and, with a single exception, only the simplest properties of Banach spaces that follow from the fact that they are complete normed vector spaces are used. This exception is Banach's open mapping theorem, which states that a bijective continuous linear mapping from one Banach space to another is bicontinuous. This theorem, trivial in finite dimensions, is stated without proof, but is used repeatedly in the book. On the other hand, the Hahn-Banach theorem is neither quoted, used, or needed. There are a few places in the text where this theorem could be used to shorten the argument; however, by avoiding the use of this result as a space-saving device, the author is able to keep the presentation both elementary and straightforward. Also, the theorems given are general in the sense that they are valid for all Banach spaces, and this again helps to maintain the elementary character of the book. As a text, the present volume could probably be used for a junior course in advanced calculus, and certainly in a senior level course. In spirit and content this book is similar to Chapters 8 and 10 of J Dieudonné's Foundations of modern analysis [1960], but the present work moves perhaps at a more comfortable and leisurely pace. The following is a partial list of the contents.

Part I is devoted to the study of calculus in Banach spaces. In Chapter 1 the basic definitions of Banach space and continuous linear and multilinear mappings are given. Some examples are presented and elementary properties established. Chapter 2 contains the definitions of the derivative (Fréchet derivative) and partial derivative. The chain rule is proved, and derivatives of some basic functions are computed. Also, given complex Banach spaces, the comparison between the $\mathbb{R}$-differentiability and $\mathbb{C}$-differentiability (holomorphic) of a function is discussed. The mean value theorem and other estimates and relations between a function and its derivative are taken up in Chapter 3. In Chapter 4 the inverse and implicit function theorems are proven, while in Chapter 5 higher derivatives are defined and Taylor's formula is established. Polynomials are studied in Chapters 6 and 7 and the relations between nth-order differentiability and local approximation by polynomials are investigated. Chapter 8 contains some classical results dealing with maxima and minima of real-valued functions. Part I closes with 32 nontrivial and interesting exercises.

Part II takes up the study of differential equations. In Chapter 1 the definitions of first-order and higher-order differential equations are given. Approximate solutions of differential equations are defined and compared, and the fundamental local existence and uniqueness theorem is proved for the case of $\Large\frac{dx}{dt}\normalsize = f(t, x)$, where $f$ satisfies a Lipschitz condition with respect to its second coordinate. In Chapter 2 linear differential equations are studied, with more detailed and specialised results given for the finite-dimensional case and the case of constant coefficients. Several distinct topics are taken up in Chapter 3 which include relations between the smoothness of the solutions and the smoothness of the vector field, equations depending on a parameter, and special results for time independent vector fields. Finally, in Chapter 4 first integrals are briefly discussed. The book closes with another set of 20 interesting and instructive exercises.

To summarise, this book provides a superb text for an undergraduate course in advanced calculus, but at the same time furnishes the reader with an excellent foundation to global and nonlinear analysis.

6.1. From the Preface by Joseph Kouneiher to a later reprint.

This book deals with differential forms and uses them to study certain local and global aspects of the differential geometry of surfaces. It is an introduction to Elie Cartan's approach to differential geometry, in which exterior differential systems and the moving frame method play an essential role. This work thus presents a modern treatment of these two subjects, including their applications to both classical and contemporary problems.

Geometric calculations and differential forms have a common origin in Grassmannian algebra; but having had different historical developments, it took time for mathematicians to recognise that they both belong to the same mathematical system

Most systems of differential equations and variational problems arising in geometry and physics admit a symmetry group. The work of Sophus Lie established that symmetric problems can be expressed in terms of group-invariant objects: differential invariants, invariant differential forms, and invariant differential operators. The Euler-Lagrange equations then inherit the symmetry group of the variational problem and can be written in terms of differential invariants.

The general group-invariant formula that allows the Euler-Lagrange equations to be established directly from the invariant form of the variational problem was first known in particular cases. Subsequently, and thanks to the work of Elie Cartan, the complete solution to this problem was determined within the more general framework of constructing a general comoving frame of an invariant form of the variational problem

Differential forms unify and simplify the calculus of variations in several variables. Students interested in this topic will obtain from his work a better conceptual understanding of the subject than that provided by conventional approaches.

Moreover, thanks to the language of differential forms that he developed during his research on infinite-dimensional Lie groups, Elie Cartan was able to solve a classification problem for all infinite-dimensional Lie groups that act locally on a two-dimensional complex space.

The importance of Cartan's theory of exterior differential forms lies in the fact that differential forms are perfectly defined objects in the context of functional substitution and pull-back (SF-PB) relative to $C^{1}$ differentiable maps going from an initial manifold of variables of dimension $M$ to a final state or an $N$-dimensional manifold. It is not necessary for the map to have an inverse (or an invertible Jacobian since the dimensions of the domain and codomain are different) since the forms and their behaviour are well-defined from a functional point of view with respect to SF-PB

This contrasts sharply with tensor analysis theory, where constraints must be imposed since mappings and their inverses, as well as differential mappings and their inverses, are required. These mappings are called diffeomorphisms and are generally used by physicists under the name of coordinate transformations. A tensor that vanishes in the final state is also zero in the initial state, but for tensors, these states must be related by diffeomorphisms. Thus, differential forms transcend tensor calculus, while still including it as a special case.

Finally, in the last chapter, the author develops E Cartan's method of moving frames to study the local differential geometry of surfaces immersed in $\mathbb{R}^{3}$, as well as the intrinsic geometry of surfaces.

In mathematics, a moving frame is an extension of the notion of an ordered basis of a vector space, often used to study the extrinsic differential geometry of continuous varieties embedded in a homogeneous space. Moving frames were first introduced by Gaston Darboux in the 19th century, for his studies of the Frenet-Serret frame of a curve embedded in Euclidean space. Subsequently, Elie Cartan's method of moving frames is based on the idea of ​​considering a moving frame adapted to the particular problem being studied. For example, given a curve in space, the first three vectors derived from the curve can generally provide a frame at one of its points. More generally, moving frames can be seen as sections of the principal bundle over the open sets $U$. Cartan's general method exploits this abstraction using the notion of Cartan connection.

The most frequently encountered case of a moving frame is that of a bundle of tangent frames on a manifold. In this case, the moving tangent frame on a manifold $M$ is composed of a collection of vector fields $X_{1}, X_{2}, ..., X_{n}$ forming a basis of the tangent space at each point of the open set.

It is always possible to define a moving frame locally, i.e., in the neighbourhood $U$ of any point $p$ in $M$; however, the existence of a global moving frame on $M$ requires topological conditions. For example, when $M$ is a circle or, more generally, a torus, this type of frame can be defined; however, this is no longer possible when $M$ is a 2-sphere. A manifold that has a global moving frame is called parallelizable. For example, latitude and longitude on the Earth's surface do not form the same moving frame whether one is at the North Pole or the South Pole.

The book's content covers both pure and applied branches of mathematics. It develops progressively according to an approach that should be particularly useful to students preparing for teaching exams, or taking differential calculus modules in a Master's program, as well as to teachers. It is also of great use to engineering students and theoretical physicists. The text is illustrated with examples, and each section includes a series of exercises of varying difficulty.

This book can be considered a reference work and a brilliant mathematical exercise.

6.2. Review by: H Osborn.
Mathematical Reviews MR0231303 (37 #6858).

The author's Calcul différentiel [1967] and the present volume are intended as textbooks for beginning and advanced calculus courses, respectively. Since differential calculus in Banach spaces is developed at length in the former book, the 108-page first chapter of the latter passes almost immediately to the definition of differential forms on open subsets of a Banach space, with values in another Banach space. Exterior differentiation is defined in the same generality, and the Poincaré lemma is established at the end of the first 35 pages. The discussion then turns to partitions of unity, changes of variables, a brief definition of differentiable manifolds, and Stokes' theorem. Chapter I closes with Frobenius' theorem, stated and proved in the expected generality of Banach spaces, and 7 pages of exercises.

The introduction of differential forms on arbitrary Banach spaces is apparently justified by its brief application in the 37-page second chapter, which develops the Euler equations for the elementary calculus of variations, with several applications to classical mechanics. The 29-page final chapter is entirely classical, however, with none of the generality of Chapter I: Élie Cartan's moving frames are defined and used to obtain several results in elementary differential geometry, concluding with the Gauss-Bonnet theorem for surfaces.

The presentation is precise and detailed, and the style is almost conversational; but no motivation is given for studying the very long Chapter I except as it occurs ex post facto in Chapters II and III. Thus, although this is clearly an elegant reference book for a serious student who already knows some advanced calculus and something about Banach spaces, only the experience of several years' use will tell whether it is really suitable for an initial exposure to advanced calculus.

7.1. Note.

This is a translation of Formes différentielles. Applications élémentaires au calcul des variations et à la théorie des courbes et des surfaces (1967).

7.2. From the Bibliographical Note.

This is the second part of a course given in 1965/66 at the Faculty of Sciences, Paris, under the title Calcul différentiel in the series Méthodes.

Translated from the original French text Formes différentielles, first published by Hermann in 1967.

The exercises are by Mme C Bullin, F Ridear and J L Verley.

7.3. From the Publisher.

Cartan's Formes Differentielles was first published in France in 1967. It was based on the world-famous teacher's experience at the Faculty of Sciences in Paris, where his reputation as an outstanding exponent of the Bourbaki school of mathematics was first established.

Addressed to second- and third-year students of mathematics, the material skilfully spans the pure and applied branches in the familiar French manner, so that the applied aspects gain in rigour while the pure mathematics loses none of its dignity. This book is equally essential as a course text, as a work of reference, or simply as a brilliant mathematical exercise.

8.1. Note.

This is a translation of Calcul différentiel (1967)

8.2. From John Moore and Dale Husemoller (editors).

Differential Calculus provides an introduction to some of the most beautiful parts of classical analysis in a modern setting, where the reader is assumed to have some familiarity with the real and complex number fields and with linear algebra at a level which is by and large covered in sophomore level mathematics courses. Often in the course of studying this book one is impressed by the masterful hand of H Cartan both in the general presentation of the subject matter and in the details of the proofs. The book is divided into two chapters. The first develops the differential calculus in Banach spaces. After an introductory section providing the requisite background on Banach spaces, the derivative is defined, and proofs are given of the two basic theorems the mean value theorem and the inverse function. The chapter proceeds with the introduction and study of higher order derivatives and a proof of Taylor's formula. It closes with a study of local maxima and minima including both necessary and sufficient conditions for the existence of such minima.

The second chapter is devoted to differential equations. Existence and uniqueness theorems for ordinary differential equations are proved. Applications of this material to linear equations and to obtaining various properties of solutions of differential equations are then given. Finally the relation between partial differential equations of the first order and ordinary differential equations is discussed.

Differential Calculus could be used for a semester junior calculus course modernising the classical advanced calculus of the junior year. A second way of using this book would be to follow its use with Cartan's companion volume, Differential Forms, for a full year course. This would be an analysis course having a geometric flavour, and providing an excellent background for further mathematical study particularly in such areas as the Theory of Lie Groups, Differential Geometry, Differentiable Manifolds, or Differential Topology.

8.3. From the Publisher of a later reprint.

This classic and long out of print text by the famous French mathematician Henri Cartan, has finally been retitled and reissued as an unabridged reprint of the Kershaw Publishing Company 1971 edition at remarkably low price for a new generation of university students and teachers. It provides a concise and beautifully written course on rigorous analysis. Unlike most similar texts, which usually develop the theory in either metric or Euclidean spaces, Cartan's text is set entirely in normed vector spaces, particularly Banach spaces. This not only allows the author to develop carefully the concepts of calculus in a setting of maximal generality, it allows him to unify both single and multivariable calculus over either the real or complex scalar fields by considering derivatives of nth orders as linear transformations. This prepares the student for the subsequent study of differentiable manifolds modelled on Banach spaces as well as graduate analysis courses, where normed spaces and their isomorphisms play a central role. More importantly, it's republication in an inexpensive edition finally makes available again the English translations of both long separated halves of Cartan's famous 1965-6 analysis course at the University of Paris: The second half has been in print for over a decade as Differential Forms , published by Dover Books. Without the first half, it has been very difficult for readers of that second half text to be prepared with the proper prerequisites as Cartan originally intended. With both texts now available at very affordable prices, the entire course can now be easily obtained and studied as it was originally intended. The book is divided into two chapters. The first develops the abstract differential calculus. After an introductory section providing the necessary background on the elements of Banach spaces, the Frechet derivative is defined, and proofs are given of the two basic theorems of differential calculus: The mean value theorem and the inverse function theorem. The chapter proceeds with the introduction and study of higher order derivatives and a proof of Taylor's formula. It closes with a study of local maxima and minima including both necessary and sufficient conditions for the existence of such minima. The second chapter is devoted to differential equations. Then the general existence and uniqueness theorems for ordinary differential equations on Banach spaces are proved. Applications of this material to linear equations and to obtaining various properties of solutions of differential equations are then given. Finally the relation between partial differential equations of the first order and ordinary differential equations is discussed. The prerequisites are rigorous first courses in calculus on the real line (elementary analysis), linear algebra on abstract vectors spaces with linear transformations and the basic definitions of topology (metric spaces, topology, etc.) A basic course in differential equations is advised as well. Together with its' sequel, Differential Calculus On Normed Spaces forms the basis for an outstanding advanced undergraduate/first year graduate analysis course in the Bourbakian French tradition of Jean Dieudonné's Foundations of Modern Analysis, but a more accessible level and much more affordable then that classic.

9.1. From the Preface.

There are three volumes. The first one contains a curriculum vitae, a "Brève Analyse des Travaux" and a list of publications, including books and seminars. In addition the volume contains all papers of H Cartan on analytic functions published before 1939. The other papers on analytic functions, e.g. those on Stein manifolds and coherent sheaves, make up the second volume. The third volume contains, with a few exceptions, all further papers of H Cartan; among them is a reproduction of exposés 2 to 11 of his 1954/55 Seminar on Eilenberg-MacLane algebras. Each volume is arranged in chronological order.

The reader should be aware that these volumes do not fully reflect H Cartan's work, a large part of which is also contained in his fifteen ENS-Seminars (1948-1964) and in his book "Homological Algebra" with S Eilenberg. In particular one cannot appreciate the importance of Cartan's contribution to sheaf theory, Stein manifolds and analytic spaces without studying his 1950/51, 1951/52 and 1953/54 Seminars.

Still, we trust that mathematicians throughout the world will welcome the availability of the "Oeuvres" of a mathematician whose writing and teaching has had such an influence on our generation.

9.2. Review by: Raymond O Wells, Jr.
Mathematical Reviews MR0540747 (81c:01031).

It is a real pleasure to see the appearance of a beautiful edition of the collected mathematical papers of Henri Cartan. The theory of functions of several complex variables has gone from its infancy with the work of Hartogs, Levi and Poincaré shortly after the turn of the century to its current role as a central field of modern mathematics, much as its predecessor, function theory in one complex variable, did in the 19th century. A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalisations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc. The major developments in the theory from 1930 to 1950 came from Cartan and his school in France, Behnke's school in Münster, and Oka in Japan. The central ideas up to that time were synthesised in Cartan's Séminaires in the early 1950's, and these were very influential to the next several generations of mathematicians. Cartan's accomplishments were broad and he influenced mathematics through his writing, his teaching, his seminars, and his students in a remarkable manner.

From the editors' preface: "There are three volumes. The first one contains a curriculum vitae, a 'Brève analyse des travaux' and a list of publications, including books and seminars. In addition the volume contains all papers of H Cartan on analytic functions published before 1939. The other papers on analytic functions, e.g., those on Stein manifolds and coherent sheaves, make up the second volume. The third volume contains, with a few exceptions, all further papers of H Cartan; among them is a reproduction of exposés 2 to 11 of his 1954/55 Seminar on Eilenberg-Mac Lane algebras. Each volume is arranged in chronological order. The reader should be aware that these volumes do not fully reflect H Cartan's work, a large part of which is also contained in his fifteen ENS-Seminars (1948-1964) and in his book Homological algebra with S. Eilenberg. In particular one cannot appreciate the importance of Cartan's contributions to sheaf theory, Stein manifolds and analytic spaces without studying his 1950/51, 1951/52 and 1953/54 Seminars. Still, we trust that mathematicians throughout the world will welcome the availability of the 'Oeuvres' of a mathematician whose writing and teaching has had such an influence on our generation."

The "Brève analyse des travaux" was written by Cartan himself in 1973, is reproduced in the first volume, and gives a very nice historical overview of his work and its interaction with the rest of contemporary mathematics from his personal point of view. It gives beautiful perspective about quite significant developments in various fields of mathematics, including function theory, algebraic topology, potential theory, and homological algebra.

These volumes are a welcome addition to the distinguished series of collected works being published by Springer-Verlag.

10.1. From the Publisher.

The Cartan-Weil correspondence is a lively introduction to part of 20th century mathematics. This book presents the correspondence, followed by 240 pages of notes and references, on the mathematical and political landscape. Readers will learn about, among other things, the birth and life of Bourbaki, the genesis, in jail, of André Weil's proof of the Riemann hypothesis on finite fields and the ferment of ideas on topology and on complex analysis which followed the invention of sheaf theory during the 1940s. They will also observe the effects of the turmoils of the century (including the rise of fascism, World War II) on mathematicians and mathematics.

10.2. Review by: Solomon Marcus.
Mathematical Reviews MR2866913.

H Cartan and A Weil were most often geographically far away from each other; Cartan was almost permanently in France, while Weil was involved in the itinerary Paris-Rome-Göttingen-Berlin-Stockholm-India (1930-1932), Marseille-Strasbourg-Finland-Sweden-France (partly in prison in the three countries), then in the USA, 1940-1945, in São Paulo (Brazil), 1945-1947, after which he moved again to the USA: Chicago, 1947-1958, and Princeton, 1958-1998. Having the same scientific start, at the École Normale Supérieure (Paris), they became friends, sharing similar research interests. Their need of interaction, in a period before the emergence of the internet and of e-mail, had been satisfied by using the traditional post. During more than 60 years, they wrote letters in which they described their questions, their attempts, their failures, their doubts, and their findings, but they were also ready to pay attention to the partner's problems and findings. The reader has the rare opportunity to look into the working laboratory of two of the most important scholars of the 20th century.

A large part of their letters is devoted to commenting on the ideas and results belonging to other mathematicians that are related to Cartan and Weil's common interests. The book contains a bibliography of more than 400 titles, including the main works of Cartan and of Weil (curiously, they have no joint paper) and an index of several hundreds of names. Most cited: Dieudonné, Chevalley, Delsarte, Leray, Godement, Laurent Schwartz, E Weil, Ehresmann, Eilenberg, de Rham, Serre, Poincaré, Hadamard, Kähler, Hopf, M Stone, Koszul, Lefschetz, Mac Lane, S Weil, Julia, Hasse, Samuel, Siegel, Lichnerowicz, and Mandelbrojt. We recognise among them some of the main members of the famous Bourbaki team, which in the 1930s inaugurated a long-term project aiming to organise the whole mathematical enterprise by means of some fundamental structures: order, algebraic and topological structures.

Structural mathematics was developed within a broader historical and cultural context, after or/and concomitantly with the emergence of structuralism in chemistry (chemical isomerism), physics (the priority of energy over the matter paradigm, the quantum revolution), linguistics with Ferdinand de Saussure, Gestalt psychology, literary theory (Russian formalism), and folkloristics (V Propp). As a matter of fact, structural mathematics was inaugurated by Évariste Galois and his concept of an algebraic group; then came Felix Klein's Erlangen Program, stressing the importance of characterising a mathematical discipline by means of some invariants with respect to some transformation groups. In the first decades of the past century, French mathematics was still underdeveloped in this direction, while the dominant trends were related to Real and Complex Analysis, Differential and Integral Equations, Differential Geometry, and Mechanics. One of the reasons for this situation was the weak interaction between French and German mathematics until the 1930s. Aware of this fact, Cartan and Weil pay great attention to the German school of Algebra emerging in the first decades of the 20th century. Weil confesses to Cartan that "he began to study 'Siegel's great memory'," and he shows high interest for the works of Behnke. Lack of interaction was an important motivation at the start of the Bourbaki Project, which proposed a new standard of mathematical rigour and stressed the structural unity of the whole mathematical enterprise. The correspondence between Cartan and Weil was an organic part of the huge laboratory which led to the long series of volumes published under the Bourbaki label. This collective work, despite all more or less reasonable criticisms, remains as one of the great mathematical events of the 20th century.

Let us explore a little in this laboratory. In the correspondence, Cartan expresses the way he alternates moments of euphoria and moments of discouragement and asks Weil to help him with a fecund idea or with an inspired question. The same Cartan is open to irony, when he sees in a tornado the rage of Bourbaki. Weil is critical about some problems considered by Cartan in connection with the concept of an `upper carapace on a compact'. Other times, Cartan is critical about the way Weil considers the so-called Poincare duality. Weil follows Cartan's indications in order to improve his proof of a theorem by de Rham. Sometimes both Cartan and Weil refer to social and political life, discussing topics such as music, human rights, and the lack of freedom to travel for some scholars in Eastern Europe. Weil expresses his admiration for the evolution to modernisation and more freedom in India according to the ideas of Gandhi and Nehru. Deeply affected by the events in 1956 in Communist Hungary, Cartan describes his incapacity, during one month, to accomplish mathematical work. With both of them moved by the dramatic events of their time, we should not wonder when, alternating with the dominant mathematical nature of their correspondence, we meet names such as Hitler, Khrouchtchev, Mussolini, Pompidou and, on p. 181, the adjective 'stalinian'. Highly interesting are the letters exchanged by Cartan and Weil with respect to the work of Élie Cartan.

A large part of this work is devoted to "Notes about the correspondence". We find here details and explanations of the conditions and of the context of the letters. Their human, cultural and historical side is fully considered. But in this respect I would like to add some information which seems to me significant about the intellectual capacity of Weil to use mathematics as a tool required by cultural anthropology, a social science apparently far away from mathematics. On p. 523, we learn that Weil has obtained his position at the University of São Paulo as recommended by the great anthropologist Claude Lévi-Strauss, who arrived in New York from São Paulo and met Weil there. Lévi-Strauss was not cultivated in mathematics, but he had a natural feeling of what is the mathematical way of thinking. As it is told on p. 523, he proposed to Weil a problem related to the rules of marriage in a specific primitive society from Australia. Weil was able to solve it and wrote a short article in this respect. Very impressed by this very clear and illuminating piece of research, perfectly intelligible by any person with a rudimentary knowledge of combinatorial thinking at the high-school level, Lévi-Strauss published it as an appendix to The elementary structures of kinship [Les structures élémentaires de la parenté, Presses Univ. France, Paris, 1949; revised edition, English translation, Beacon Press, Boston, MA, 1969], a book which became a classical reference in the field. But what is not said by the editor is the fact that, with this text, Weil created a new branch in the field of the sociology of kinship relations, called "The Mathematics of Kinship Relations", a domain which is still very active.

Many other aspects of this correspondence deserve to be mentioned: (a) We may capture signs of the evolution from the traditional Italian school of Algebraic Geometry to the modern one when both Cartan and Weil give a negative reaction to a long article by Francesco Severi. (b) Attention is paid to the birth of Mathematical Reviews (1940), to which both of them contributed. (c) This correspondence begins in a moment when joint works in mathematics were not at all usual. It became almost a rule in the second half of the 20th century and Bourbaki remains as a pioneer of collective mathematical enterprise. (d) Frequently, in their correspondence, Cartan and Weil communicate to each other some fresh new results. This fact reminds us of the situation in the past centuries, when letters were the main way to convey mathematical ideas and results. (e) The middle of the 20th century, which is the most interesting period of this correspondence, is characterised by a big change: if in the first half of the past century the centre of the mathematical life was Western Europe, with the triangle Paris (Sorbonne)-Göttingen-Rome in the main attention, in the second half of the century the USA became more and more the main scene of mathematical life and this situation is clearly mirrored in this volume. As a matter of fact, after 1940 the French Weil becomes the American Weil and some other members (or associates) of the Bourbaki group are American. The Bourbaki treatise has been translated in English, although the Russian translation came before the English version; but let us recall that before the middle of the past century the interaction between European and American mathematics was rather weak. (f) Bourbaki changed the basic mathematical terminology, promoting a metaphorical one. This fact is frequently reflected in the correspondence between Cartan and Weil. Few mathematicians are aware of the origin in Bourbaki of some notations which are now generally accepted in Algebra, Analysis and Topology.

To conclude, I think that this work became possible because some sociological and cultural conditions were satisfied. First of all, the descendants and the relatives and friends of both Cartan and Weil were very cooperative and aware of their duty to honour their memory. France is a country with a long tradition of respect for cultural heritage. Many people and institutions, in France and abroad, gave their support. The editor Michèle Audin deserves high appreciation for this work; a translation in English would be very useful.

10.3. Review by: Javier Fresán.
European Mathematical Society Newsletter (June 2012), 58-60.

This book assembles more than 200 letters exchanged by Henri Cartan and André Weil from November 1928 to May 1991. Most of them were discovered a few years ago within the archives of Cartan, who does not seem to have thrown away a single paper in his life. It would be hard to imagine a better editor for this correspondence than Michèle Audin, an expert, among other things, on the history of French and German mathematicians during the World Wars and the interbellum. The exquisite research she has carried out becomes evident from the first page. In particular, her extensive notes at the end of the volume are not reduced to a mere identification of the various characters and situations to which the letters refer; on the contrary, they "tell another story", in the same way that the commentaries added by Weil to his collected works form an independent book. One can find there, just to mention a few examples: a long letter in which a very young Weil displays all his mathematical knowledge; a chronology of the Cartan seminar through Serre's memories; and a thorough reconstruction of the anticommunism hysteria surrounding the ICM 1950, which part of the French delegation was planning to boycott if Hadamard and Schwartz did not get their visas in time. Several documents from the recently declassified files of Bourbaki have also been included.

Let it be said from the beginning that this correspondence is quite different in style from the one maintained, partly at the same time, by Grothendieck and Serre, of which it could be reminiscent at first sight. While the main topic is of course mathematics, it is not the only one: as Cartan and Weil were close friends and founding fathers of Bourbaki, many letters address practical problems regarding the organisation of the group and questions of a more personal nature (such as family holidays, health issues and music). A particularly sad leitmotif is Weil's recurring desire to find an academic position in France, for instance when Lebesgue retired from his chair at the Collège de France. Despite the great deal of time and energy Cartan devoted to supporting his friend, all his attempts were frustrated by political resentment and the direct application of Weil's own law, "first-rate people at-tract other first-rate people but second-rate people tend to hire third-raters and third-rate people fifth-raters". Back to mathematics, it is thrilling to discover how the slowly emerging notion of "cohomology with variable coefficient domains" had already led, in the late 1940s, to a perfectly modern definition of spectral sequences. Young and not-so-young readers will probably smile at how breakthroughs such as the Steenrood operations and the Kodaira embedding theorem "every variety of Hodge is a projectile variety" were disseminated before the arXiv!

Bourbaki

It could be seen as disappointing not to find any scoop here on the birth of Bourbaki. But this is not surprising: at that time, Cartan and Weil were colleagues at the University of Strasbourg so why should they exchange letters when they could speak in person? The first time Bourbaki is mentioned, on 29 May 1939, is just to say, "Bourbaki is becoming very popular everywhere: in Cambridge he is now the most talked-about mathematician. I have heard that Chevalley did a lot of propaganda at Princeton." This shows that it was not yet the secret society it was going to become in the following years; in contrast, Weil was angry to learn on 4 May 1955 that Saunders Mac Lane had delivered a public speech at New York University in which he described himself as a "fellow-traveller" of Bourbaki. Thanks to the letters, other elements of the legend can be put into historical context. For instance, one confirms that retirement at 50 was not a rule until the moment that Weil reached this age and wrote to Cartan, "The best service that the founding members can currently render to Bourbaki is to disappear gradually, but within a finite timeframe." If something is to be taken from the correspondence, it is that our protagonists always had Bourbaki in mind. Three early letters show Weil's insistence on replacing the term "ensemble vasculaire" by "ordonné filtrant". Far from being an exception, that was the general trend. Even the smallest typographical details were discussed at length; nevertheless, Weil was not unaware of the risks of this way of working, as the following extract from Bourbaki's bulletin La Tribu shows: "we cannot continue to waste all our time on trivialities. When the content of a chapter becomes stable, there is no longer a need for a plenary congress to discuss the details." Taking into account the method, the scarcity of paper, and the slowness of the postal service, it can only be regarded as a miracle that Bourbaki survived the war. A letter not to be missed, dated 19 July 1946, is the one in which Cartan suggests, following Chevalley, that modules could be expelled from Bourbaki's Algebra: "If we limit ourselves to vector spaces, the exposition is much more aesthetically pleasing, we undoubtedly avoid cumbersome passages, and we facilitate the task of the majority of readers who, obviously, will only be interested in vector spaces. It goes without saying that this sacrifice can only be made if the interest of modules, in the rest of the Algebra, is to be sufficiently limited so that we can, if not do without them entirely, at least relegate them to the precise place where they will be needed;" this follows a choleric five-page answer by Weil which definitively closed the issue.

The Weil conjectures

Another set of letters concerns the proof of the Riemann hypothesis for curves over finite fields during the Spring of 1940. In those days, Weil was imprisoned in Rouen after what he would later call "a disagreement with the French authorities on the subject of my military obligations". He did not waste this opportunity to work "sans souci extérieur", as Cartan put it: besides proof-reading his first book and reconstructing a report on integration for Bourbaki, which had been confiscated by the Finnish police, Weil continued thinking about zeta functions. On 26 March, he writes to Cartan, "I believe I am touching upon very important results concerning the function of fields of algebraic functions." He then insists on the urgency of getting the answer to a question he has already asked his friend: 'What is the number of $n$-torsion points of the Jacobian of a curve of genus $g$ over a finite field?' This was needed for the "important" lemma on which his whole argument to prove the Riemann hypothesis relied. On 8 April, the same day that he wrote an illuminating letter to his sister, Weil announced to Cartan that he had submitted a note to the Académie des sciences: "More seriously, I sent the note without waiting to prove the fundamental lemma; but I now have enough clarity on these matters to take the risk. I have never written anything, and I have almost never seen anything, that reaches such a high degree of concentration as this note. Hasse has nothing left to do but hang himself, for I resolve in it (subject to my lemma) all the main problems of the theory." As Weil imagined, German mathematicians did not take long to react, initiating a true "war of reviews"; however, the correspondence gives no clue about his feelings regarding the accusation of "unfair play". In 1942, Weil already knew how to prove the lemmas but the complete argument would only be published "eight years and more than five hundred pages later"; some letters (starting at p. 97) treat this unusual delay, which is partly due to Weil's refusal to split one of his memoirs into several articles. Somewhat more surprisingly, no mention is to be found in the remaining correspondence either to Weil's paper Number of solutions of equations in finite fields or to the long-range programme culminating in the proof of the conjectures stated there. To remedy this, Audin has included a fascinating letter from Weil to Delsarte, dated 13 September 1948, in which he sketches the proof of his conjectures for Fermat hypersurfaces and relates the Ramanujan conjecture to this circle of ideas.

Algebraic topology

A less expected chapter of the correspondence deals with ideas on topology and complex analysis around the invention of sheaf theory. Let us recall that Cartan was the first person to unravel the obscure papers by "l'illustre Leray" and to embark, through his seminar, the new, brilliant generation upon the search for applications. On his side, Weil was perfectly up to date with the progress on topology, as this was the field he had chosen to collaborate with the recently created Mathematical Reviews. Of course, the correspondence contains the already published letter in which Weil explains how to prove De Rham's theorem on duality between singular chains and differential forms; but this is now completed with a second letter in the same vein. Cartan's manuscript margin notes show that he had studied both texts in detail: in particular, he asks how to define, in the topological setting, "the cohomology ring (i.e. the product operation)," which should correspond to the wedge product of differential forms. This was at the origin of his theory of "carapaces", an alternative to Leray's "couvertures", which appears on stage for the first time on 5 February 1947. Naturally reserved, Cartan was really enthusiastic about the power of this new notion: "By reflecting on it, you will gradually perceive for yourself the scope of this new theory, which encompasses, while considerably simplifying them, all the known, seemingly so divergent, aspects of algebraic topology." Even if Weil remained sceptical for a long time, this did not stop him from encouraging Cartan to pursue his research. Another interesting exchange was intended to help his friend prepare his ICM talk Problèmes globaux dans la théorie des fonctions analytiques de plusieurs variables complexes; several letters concern the second Cousin problem and the difference between topological and analytically trivial fibre bundles.

Needless to say, this precious document deserves much more careful analysis. Just to mention an aspect not treated in the preceding sections, the beautiful letter dated 15 June 1984 leaves no doubt as to how highly Weil thought of his friend's father Élie Cartan, one of the secondary characters of the correspondence. My only aim here has been to draw attention to some of the passages I liked the most. Find your own!