1.1. Review by: G Doetsch.
Zbl 0055.34203

J von Neumann has established a very general spectral theorem that is analogous to the usual spectral theorem for Hermitian matrices. This is first applied to developments related to Carleman's kernels, which gives rise to the generalization of a theorem by F J Mautner. This in turn yields a developmental theorem for self-adjoint elliptic differential operators that generalizes known results of H Weyl and M H Stone.

2.1 From the Preface.

The purpose of these lectures is to give proofs of the basic existence theorems connected with Cauchy's problem for hyperbolic equations. This has been done twice before, first by Petrowsky and later by Leray in his Princeton lectures, but the subject presents considerable technical difficulties which may justify another treatment. Our basic tool will be the energy integrals associated with separating hyperbolic operators introduced by Leray, The main new feature is that Cauchy-Kovalevskaya's theorem is replaced by a certain inequality which simplifies the theory and also sharpens the results. This approach is not new; it has been used by Friedrichs, Lax and Ladyzenskaja in special cases. Another novelty is the use of partial adjoints which elucidates Cauchy's problem for distributions.

3.1. From the Preface.

Trying to make mathematics understandable to the general public is a very difficult task. The writer has to take into account that his reader has very little patience with unfamiliar concepts and intricate logic and this means that large parts of mathematics are out of bounds. When planning this book, I set myself an easier goal. I wrote it for those who already know some mathematics, in particular those who study the subject the first year after high school. Its purpose is to provide a historical, scientific, and cultural frame for the parts of mathematics that meet the beginning student. Nine chapters ranging from number theory to applications are devoted to this program. Each one starts with a historical introduction, continues with a tight but complete account of some basic facts and proceeds to look at the present state of affairs including, if possible, some recent piece of research. Most of them end with one or two passages from historical mathematical papers, translated into English and edited so as to be understandable. Sometimes the reader is referred back to earlier parts of the text, but the various chapters are to a large extent independent of each other. A reader who gets stuck in the middle of a chapter can still read large parts of the others. It should be said, however, that the book is not meant to be read straight through.

3.2. Review by: K E Hirst.
Mathematical Reviews MR0462823 (57 #2796).

The aim of this book is to introduce beginning undergraduates to an overall view of mathematics, to give them some kind of framework within which to set their detailed undergraduate studies. Much of the book requires a mathematical sophistication in advance of this, however, for in attempting to avoid a large amount of detail the author gives very concise proofs, and the results that are stated without proof are in many cases preceded by definitions of mathematical structures and concepts that can by their nature communicate little of the nature of an area of mathematics to the uninitiated. ... Books of this kind are very difficult to write, and very easy to be critical of. There are undoubtedly many fascinating things in the book, but I feel that in its almost exclusive attention to pure mathematics it can at best give the general (mathematically mature) reader a rather one-sided view of the subject. Almost every professional mathematician will find large gaps in the book, but that is bound to be so in an exposition of this kind. It is emphatically not a book to learn mathematics from. It is an honest attempt to come to grips with the immensely difficult problem of telling people unacquainted with mathematical theories what those theories are about.

3.3. Review by: Victor Bryant.
The Mathematical Gazette 62 (421) (1978), 217.

The aim of this book is to provide "a historical, scientific, and cultural frame for the parts of mathematics that meet the beginning student." To achieve this aim nine of the chapters each have a main theme (including, for example, "number theory," "geometry and linear algebra" and "differentiation") and within each of these chapters there is a sketch of the historical development, an account of what the author regards as the basic facts of the subject, and a glimpse of the present state of affairs. To present the whole of mainstream mathematics in this way (more a full parade than a chance encounter) is a major achievement. The end-product is a pleasure to read and certainly for someone like myself, who graduated some years ago and has taught several aspects of the subject since, it is a pleasant reminder of the roots of the various branches of modern mathematics and of the links between them. However, the author wrote this book for students in their "first year after high school," which I interpret as our first year undergraduates. This aim is, in my opinion, hopelessly optimistic, as only an outstanding first year undergraduate would gain any insight from reading this book; indeed most would fail to understand all but a handful of the concepts. Most people can only get a perspective of a subject (especially when presented in as much detail as this book is) by looking back on it, rather than forward.

4.1. From the Historical Introduction.

The theory of wave propagation started with Huygen's theory of wave front sets as envelopes of elementary waves. Its first success was the proper explanation of the propagation of light in refracting media. Its modern successor is the theory of boundary problems for hyperbolic systems of partial differential equations. The development which led to this theory is a story of the search for proper mathematical tools.

4.2. Review by: Vesselin M Petkov.
Mathematical Reviews MR0905058 (88k:35002).

This book presents an application of the ideas of microlocal analysis to the examination of the singularities of solutions to linear hyperbolic equations. Chapter 1 is devoted to the construction of the fundamental solution of hyperbolic operators with constant coefficients. Here are discussed propagation cones, general conical refraction and the Herglotz-Petrovski-
formula. In Chapters 2 and 3 the basic facts from microlocal analysis including wave front sets, Fourier integral operators and the calculus for pseudodifferential operators are presented. Chapter 4 contains some material from symplectic geometry which is necessary for the exposition. In Chapter 5 the author proposes a new simple construction of a global parametrix for the fundamental solution of a first-order pseudodifferential operator. This construction leads to some oscillatory integrals paired in a suitable way. Chapters 6 and 7 contain a careful analysis of these paired oscillatory integrals. Here are defined sharp and diffuse fronts, Petrovskii chains and cycles, and the singularities of such integrals are studied. The book is very well written, the exposition is clear and combined with historical remarks. These lecture notes are an excellent introduction to the analysis of singularities for hyperbolic operators.

5.1. From the Preface.

The aim of this book is to teach the reader the topics in algebra which are useful in the study of computer science. In a clear, concise style, the author present the basic algebraic structures, and their applications to such topics as the finite Fourier transform, coding, complexity, and automata theory. The book can also be read profitably as a course in applied algebra for mathematics students.

5.2. Review by: Boris M Schein.
Mathematical Reviews MR0953965 (90b:00003).

This is a textbook that introduces the reader to number theory, abstract algebra (including modules, rings and fields, groups, Boolean algebra and automata) and to various applications of these parts of mathematics to computer science. The reader is not expected to have any previous exposure to abstract algebra except familiarity with the elements of linear algebra and a certain mathematical maturity. Some parts of basic theory are left to the reader as exercises (referred to as proven results). The book also contains many ordinary exercises.

5.3. Review by: Anne C Baker.
The Mathematical Gazette 74 (468) (1990), 187-188.

In 198 pages this book contains a wealth of algebra with some applications to computer science. I found the applications both interesting and revealing. However I do wonder at whom the book was aimed. The algebraic content covered is comprehensive but very condensed. Any reader with a computer science background could well falter at the pace at which the algebra and number theory is presented. The book contains a very good summary of the pure mathematical concepts required for the applications but if this is envisaged to be an introduction to this material then the book fails. The pure mathematics is clearly and concisely presented but, for me needs to be read in very small doses. However a good full reference text is provided for the pure mathematician interested in computer science. My only reservation is that it may be too full. The applications themselves are well described The topics covered include groups, modules, rings, algebraic and finite fields, monoids, Boolean algebra, algebraic numbers, primes, pseudo-random numbers. The main applications are to the fast Fourier transform, complexity, shift registers and coding theory, automata, languages.

6.1. From the Preface (English translation).

The purpose of this book is to document mathematical research in Sweden before 1950: what was done, who did it and, partly, why it was done. So far only the work of the Swedish eighteenth century mathematician Samuel Klingenstierna has been documented in this way by the physicist Carl W Oseen and the librarian Hildebrand Hildebrandsson ( 1919). With the exception of Mitteg-Leffler I have been restrictive with biographical detail. My purpose has been to write about the work of mathematicians for readers interested in mathematics.

The year 1950 is a natural limit. The ensuing forty-five years have given a historical perspective and the end of the war was also the end of a university organization in Sweden which had lasted for 8 hundred years. During that time a university professor was often the only one responsible for the teaching and examination in his subject . In addition he was expected to do research and publish new results. Sometimes he had the help of a docent on a six-year scholarship. In practice, these two categories of people were the only ones who had sufficient motivation and time for research. Because the system promoted the most successful docents to professors it is inevitable that the greater part of my book is about the work of professors of mathematics. But almost all theses in the subject from 1890 to 1950 are mentioned and the most important ones treated in more detail.

In some cases which I have thought interesting, I have gone outside the circle of mathematicians, but applied mathematics in general is outside my scope. In particular, most mathematical physicists are excluded. Probability is mentioned, but only in connection with the mathematicians who have written in this field.

All mathematical papers are of course not mentioned. Those chosen for commentary have been interesting for at least one of the following reasons. One is quality, another is a wish to give a fuller mathematical profile of the most interesting mathematicians. Sometimes a paper has been selected because its background or mistakes have been more interesting than the paper itself.

In order to break a monotonous sequence of remarks and comments I have sometimes sketched the social setting and, when possible, challenged the reader by reproducing or at least sketching some proofs.

The decisive event for the position of mathematics in Sweden happened in the 1880's when Gösta Mittag-Leffler, the first professor to be nominated at Stockholm University, gathered a circle of young mathematicians and founded an international mathematical journal, the Acta Mathematica. Since then Sweden has not been lacking in good mathematicians.

I dedicate my book to Swedish mathematicians, the circle in which I spent my active professional life.

6.2. Review by: Thomas Archibald.
Mathematical Reviews MR1317098 (96a:01024).

This book provides an able survey of the very rich mathematical activity in Sweden from the early eighteenth century to 1950. It is divided into sixteen chapters, focussing on individual mathematicians and their mathematics, on mathematical topics, and on the schools at Stockholm, Uppsala and Lund. There is a great deal of biographical and bibliographical information both on well-known figures and on relatively obscure individuals (lists of dissertations were particularly welcome). In addition, the concise discussions of the mathematical work (usually in modern notation) often assess both origins and influences in a thought-provoking way. The bulk of the book is devoted to the twentieth century. The overall effect is not unlike that of Felix Klein's study of mathematics in the nineteenth century: selective and opinionated, but invaluable as a reference work and a source of insight.

7.1. From the Publisher.

This book is a collection of small essays containing the history and the proofs of some important and interesting theorems of analysis and partial differential operators in this century. Most of the results in the book are associated with Swedish mathematicians. Also included are the Tarski-Seidenberg theorem and Wiener's classical results in harmonic analysis and a delightful essay on the impact of distributions in analysis. All mathematical points are fully explained, but some require a certain mature understanding from the reader. This book is a well-written, simple work that offers full mathematical treatment, along with insight and fresh points of view.

7.2. Review by: Michael von Rentein.
Mathematical Reviews MR1469493 (98m:01023).

In the booklet under review the author, a leading mathematician himself, considers 12 fundamental achievements in the history of analysis in the 20th century. The first five chapters are devoted, respectively, to Picard's great theorem, Holmgren's uniqueness theorem, the Phragmén-Lindelöf principle, Nevanlinna theory and the Riesz-Thorin interpolation theorem. Chapter 6 on Wiener's Tauberian theorem describes nicely the course from Wiener's classical Annals paper (1932), via Gel'fand's theory of normed rings, to Fourier analysis on locally compact abelian groups. The six last chapters deal with the Tarsky-Seidenberg theorem, intrinsic hyperbolicity, hypoellipticity, the Dirichlet problem and Gårding's inequality, a sharp form of the latter inequality and the impact of distributions in analysis. The mathematical details are well presented and the proofs to the main theorems are given.

7.3. Review by: Richard Beals.
Bull. Amer. Math. Soc. (N.S.) 35 (2) (1998), 157-160.

This lively book is a guided tour through some of the highlights of twentieth-century analysis. It has the strengths and weaknesses of a series of lectures: it is personal, readable, uneven, and occasionally repetitive rather than anonymous and organized to perfection. All three of the standard subunits of modern analysis - complex, real, and functional - are represented, the latter two especially in the context of PDEs. ... The author is an originator of the modern approach to PDE through systematic use of the Fourier transform and functional analysis. ... The book covers a wide range of material, much of which should be of interest to any student of analysis. It allows one to see much of the terrain from the point of view of a major participant as the threads of hard and soft analysis were woven together in the latter part of the century. This is clearly not a textbook, but it is far from a history - one learns a bit about who taught at which Swedish university, who was whose student, and what might have motivated some of the mathematics, but after a promise to explain the origin of the "unnecessarily complicated name" of Wiener's Tauberian theorem comes a paragraph that mentions asymptotics but neither Abel nor Tauber. However, there are interesting tidbits, like the famous thesis which, even by the time of the public defence, had not been read by the advisor. For use, rather than perusal, the book can be recommended, for example, to a graduate student seminar. By filling in the details and doing a bit more reading in the references as necessary, one can pick up a number of the gems of analysis without having to mine whole seams.

7.4. Review by: Bruce Kellogg and Vidar Thomee.
SIAM Review 40 (4) (1998), 1007-1008.

This delightful little book (only 88 pages) is a somewhat augmented collection of lectures given during a visit to China in 1994. The 12 lectures survey some of the most important results in mathematical analysis of our century, with emphasis on matters close to the heart of the author. Much of the material is concerned with the development of the modern theory of general linear partial differential equations, in which the author has played a leading role, and also with the tools from mathematical analysis which have had a particularly decisive effect on this development, such as distribution theory, the Riesz-Thorin interpolation theorem, the Tarski-Seidenberg theorem, etc. A pervading feature is the presence of historical comments and very personal recollections from someone who was actually there and very much part of the development. As is natural both because of the culture the author comes from and the importance of the contributions from his part of the world, work of Swedish mathematicians play a major role; the following list of people quoted gives a flavour of this: Beurling, Carleman, Carlsson, Fredholm, Holmgren, Hörmander, Mittag-Leffler, and M Riesz, and one could add Ahlfors, Lindelöf, and Nevanlinna from neighbouring Finland. ... Most personal are perhaps the accounts of the areas in which Lars Gårding made his own major contributions, such as the lecture on hyperbolic operators with its elegant study of the corresponding polynomials and that on Dirichlet's problem, where the historical development culminating in Gårding's inequality is lucidly expounded. Also fascinating is the final lecture in which the author recounts, along with a description of the theory, his personal awakening to the importance of distributions. The style is elegant and lively, and the book provides exciting and instructive reading for anyone who is interested in the pure mathematics behind and often exploited in the analysis of many of the applied problems that are relevant to the readers of SIAM Review. The reviewers are nostalgically reminded of privileged times as students of their warmly admired teacher.

8.1. From the Publisher.

This book is about mathematics in Sweden between 1630 and 1950 - from S Klingenstierna to M Riesz, T Carleman, and A Beurling. It tells the story of how continental mathematics came to Sweden, how it was received, and how it inspired new results. The book contains a biography of Gosta Mittag-Leffler, the father of Swedish mathematics, who introduced the Weierstrassian theory of analytic functions and dominated a golden age from 1880 to 1910. Important results are analysed and reproved in modern notation, with explanations of their relations to mathematics at the time. The book treats Backlund transformations, Mittag-Leffler's theorem, the Phragmen-Lindelof theorem and Carleman's contributions to the spectral theorem, quantum mechanics, and the asymptotics of eigenvalues and eigenfunctions. Other important features include sketches of personalities and university life. This book presents the first thorough treatment of mathematics in Sweden. It discusses the work of the great mathematicians and the development of mathematics throughout Europe. It also brings the mathematics of an era to life in an informative and highly readable way.

8.2. From the Preface.

The Preface is an English translation of the Preface to the Swedish edition. See 6.1 above.

8.3. Review by: David E Rowe.
Isis 89 (3) (1998), 554-555.

Written by an eminent mathematician for fellow mathematicians, this survey represents the first sustained effort to document the historical development of higher mathematics in Sweden. As the redundancy in the book's title suggests, Lars Gårding's principal concern is to chronicle Swedish contributions to the history of mathematical knowledge while giving snapshots of the careers of a few leading figures. Following two brief chapters describing mathematics at the universities in Uppsala and Lund from 1700 to 1850, Gårding's story begins to gather momentum. Mathematical ideas largely dictate the organization of the ensuing narrative, although the chapters are often structured around activities at one of the three Swedish universities, a format that produces considerable flip-flopping in the chronology. Thus Chapter 5, for example, which deals with Victor Backlund's work on partial differential equations, is sandwiched between chapters on general developments at Uppsala and Lund between 1850 and 1900. Although often anachronistic, Gårding's explanations of the many mathematical works he describes are usually quite insightful, and in this chapter he gives a clear and concise account of Backlund's contributions within their immediate intellectual context, stressing the importance of Sophus Lie's theory of contact transformations. The surrounding chapters, however, scarcely deserve to be called institutional history. Readers who wish to gain a sense of the mathematical research traditions associated with Sweden's institutions of higher education will find plenty of relevant information in Gårding's book, but there is almost nothing about the forces and issues that animated that research. ... Although Gårding's undertaking must be regarded by historians of mathematics as both welcome and long overdue, the outcome regrettably turns out to be yet another example of that familiar genre in which mathematical worlds appear as at once self-contained and self-sustaining. The late Richard Westfall lamented a similar characteristic in S Chandrasekhar's book on Newton's Principia, which he called "disturbing" and even hostile toward historians of science. But should we expect mathematicians who are oblivious to all the complexities that confront the serious historian to forgo the intellectual pleasures of communing with the past in perfect harmony? In the present case, I find Gårding's lack of historical documentation and perspective far less disturbing than the realization that a book with such glaring weaknesses was approved for publication by an editorial board whose members include a number of well-known historians of mathematics.

8.4. Review by: Nick Lord.
The Mathematical Gazette 84 (499) (2000), 164-165.

In this book, which he dedicates to Swedish mathematicians, the distinguished analyst Lars Gårding gives an in-depth chronological account of the history of mathematics in Sweden. This is largely a tale of three university centres Uppsala (founded in 1477), Lund (1668) and Stockholm (1880); (moreover, with just half-a-dozen mathematics professors between them even early in this century, it is a story largely driven by the talents and charisma of individuals. Gårding only cites the work of two pre-nineteenth century mathematicians (Klingenstierna and Bring) and the bulk of the book centres on the period 1860 to 1950 - from the birth of modern Sweden to the post-war reorganisation of its universities. He supplies a crisp, urbane commentary with neat summaries of background mathematical themes (in modern gloss) and an insider's sympathy for peculiarities of Swedish academic life. One of these concerns the perils of appointments by open committees: we learn that in 1873, Björling was preferred to Backlund (of transformations fame) and that, in 1923, the experts were unable to decide between the merits of Marcel Riesz and one Nils Zeilon! Although Gårding's primary aim is 'to write about the work of mathematicians for readers interested in mathematics', he fleshes out the mathematics with historical and biographical details. He is not afraid to express strong opinions and his witty one-liners, together with anecdotal quotations from archival sources, and the 40 photographs at the end of the text serve to bring the pen-portraits to life. ... But, as Gårding rather elegiacally reflects in a postscript, the attraction of a book such as this is not so much in the rehearsing of those names which Posterity sanctifies but in the dusting-off of those which Time has forgotten: who now has heard of Björling, Dillner, Falk, Malmsten, Wiman, or Gullstand (who won the 1911 Nobel Prize for Medicine)? Indeed, to share a personal reminiscence, the name Edvard Phragmén rang a rusty bell with me from the Phragmén-Lindelöf principle (an extension of the maximum modulus theorem to sectors). He now emerges from the shadows first as the eagle-eyed proof-reader who detected Poincaré's serious mistake in his initial submission for the King Oscar II Prize (the rectification of which led Poincaré to his intimation of the possibility of chaotic behaviour of solutions to the three body problem). He then succeeded Kovalevski as professor at Stockholm in 1892 but (as Erdös would have put it) he 'died' in 1903 to become a highly successful chief inspector of insurance! I found this book full of such surprises and fresh insights: I can warmly recommend both it, and the preceding twelve volumes of the joint AMS/LMS History of Mathematics series in which it takes it place.

8.5. European Mathematical Society Newsletter.

The text brings an expert overview of the development of mathematics, presented in a master style that reflects a deep insight into the subject. Thus the readers learn important facts from the history of mathematics and extend their previous knowledge in fields that do not exactly overlap their own specialties. This book on mathematics and mathematicians is strongly recommended to anybody who likes mathematics and its history.