Book Description

Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. It is no coincidence that the fundamental science of numbers has come to be known as the "Queen of Mathematics." Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory.

Customer Reviews (1)

Respectable and enjoyable guide to pre-Gaussian number theory
When someone like Weil sets out to write a history of number theory it is destined to be the standard reference for decades to come. But this is not only an authoritative reference everyone loves to cite--it is also delightfully readable. It is not a substitute for a textbook (although Weil hints at this possibility is the preface), but even for readers with only a modest background in number theory this book will be a source of insight and joy.

Chapter 1 "Protohistory" treats briefly some of the scattered pre-Fermat attempts, which helped form the Diophantine tradition of what would constitute the staple problems of number theory--Pythagorean triples, sums of squares, Pell's equation, such things. These seeds blossomed in the hands of Fermat (chapter 2), with whom we start to see the formation of a coherent theory of numbers with some basic tools: infinite descent, modulo arguments, a precursor of elliptic curve arithmetic, etc. Fermat rarely wrote things down properly, and Euler (chapter 3) had to work hard to prove his theorems and conjectures, in the process adding some ideas of his own (the group theoretic core of modulo arithmetic and Z/pZ, auxiliary functions such as the phi function, etc.). Euler's further investigations along these lines also left many valuable ideas for future mathematicians such as the crystallisation of the importance of quadratic forms (taken up by Lagrange, chaper 4; later perfected by Gauss) and the statement of the law of quadratic reciprocity (taken up by Legendre, chapter 4; later proved in full by Gauss). Also highly decisive for the future development of number theory was Euler's bringing in of analytic ideas into number theory, in particular elliptic integrals (whose deep importance was later revealed by Jacobi) and the zeta function and L-series (whose deep importance was later revealed by Dirichlet and Riemann). ... Read more

Customer Reviews (1)

A warning to beginners
Experts find this a very good book, and I rate it on their advice. But others need to understand that Weil is making a bit of a joke with the title. This book is "basic" in the sense that it proves the theorems that Weil feels organize and clarify all of number theory--the "basic" theorems in that sense. It is an introduction to class field theory.

As Weil says at the start of the book, it has few prerequisites in algebra or number theory, except that the reader is presumed familiar with the standard theorems on locally compact Abelian groups, and Pontryagin duality and Haar measures on those groups. This part is not a joke.

If you want to really understand class field theory this may be a good book. (I am reliably told it is.) But Weil deliberately avoids using many ideas that are now standard: geometric ideas such as group schemes, and especially cohomological methods.

Beginners studying algebraic numbers do not need this book. Weil recommends Hecke ALGEBRAIC NUMBERS for such readers, and that is a terrific book. To learn class field theory today you'd probably do better with and Cassels and Frohlich ALGEBRAIC NUMBER THEORY, which Weil also recommends in a note to the second edition of this book. ... Read more

Book Description
The author, a mathematician whose horizons have never been limited to mathematics, recalls a career that led him to numerous continents: to Italy and Germany first of all; then to India where he lived and taught at a critical time in the history of that country, and where he met with Gandhi and Jawaharlal Nehru; to Russia when Stalinism seemed to be waning only to then rise up again with increased ferocity; to Princeton, the modern "clearing house" of mathematical ideas, called at times a mathematician's paradise; to a prison in Finland where, taken for a Soviet spy, he narrowly escaped execution; to France, where he was convicted for dodging his military obligations (the draft) and where, in the prison of Rouen, he had time to write one of his best mathematical works; to England, where he lived through the Battle of London before returning to France and then to the United States; and finally to Brasil, scene of the last of his vicissitudes, before returning permanently to the United States. Through these often picturesque episodes, the destiny of a mathematician is unfolded, of which perhaps the most salient event was his participation in the foundation of the Bourbaki Group, an auteur collectif , of a treatise that has long since become a classic.

Translated from the French by Jennifer Gage, with assistance of the author. ... Read more

Customer Reviews (5)

The recollections of a great mathematician
The title is a little misleading since there is not much math in the book, but a lot of personal stories about the intellectual life and travels of A. Weil. Through them we can glimpse his personality and thus the book will be interesting to mathematicians and historians of mathematics. The reader can detect a completely undogmatic mind, skeptical about justice and politics and with a rather ironic sense of humour. His stay in jail for not reporting for duty at the start of WWII was one of his most productive periods.

He recalls, but he is not able to give a concrete date, the day when H. Cartan and he founded Bourbaki. One of the funniest anecdotes is when Cartan receives a call from a Greek whose name is Bourbaki thinking it is a joke, but they become friends and he is even invited to some of their meetings.

A. Weil was not only an outstanding mathematician but a man of a wide culture and a polyglot. He studied sanskrit to read the great books of Indian literature and spent a couple of years in India. Another interesting story is when he is taken by a Russian spy in Finland and he is saved in extremis from execution by a chance meeting of Nevalinna with the chief of police.

He eventually moved to the US, although the first years must have been very frustrating teaching at second rates colleges where when he provided a proof the students would ask: "Is it going to be in the exam?"

The book ends in the fifties when he is appointed a professor at the University of Chicago after a stay in Sao Paulo.

To sum up, a fascinating personality that had a fascinating although not always easy life.

Light reading!
There is only a small number of autobiographies by mathematicians. Andre Weil was a giant in math. His autobio, written late in life, is fun to read. Weil has strong opinions that may perhaps not appeal to all. Even so, the book light reading, agreeing or not; and it fun too. For me it was a page-turner. To others perhaps a little pompous. Judge for yourself. While perhaps self-absorbed, I think Weil in his autobio gives personal and fresh insight into the tumultuous period in history, between the two World Wars in Europe, as it relates to math. The main part of the book covers Weil's life before he came to the US.
Weil had a monumental impact on math, and he also wrote some lovely history of math books, --number theory; and then of course some specialized books, that are corner stones in math, but not especially easy to read, at least for beginners. But Andre Weil is a central figure in math. His younger sister Simone Weil was an author and philosopher, and a political activist on the left in French politics in the 1930ties. She died young.

Light reading!
There is only a small number of autobiographies by mathematicians. Andre Weil was a giant in math. His autobio, written late in life, is fun to read. Weil has strong opinions that may perhaps not appeal to all. Even so, the book light reading, agreeing or not; and it fun too. For me it was a page-turner. To others perhaps a little pompous. Judge for yourself. While perhaps self-absorbed, I think Weil in his autobio gives personal and fresh insight into the tumultuous period in history, between the two World Wars in Europe, as it relates to math. The main part of the book covers Weil's life before he came to the US.
Weil had a monumental impact on math, and he also wrote some lovely history of math books, --number theory;and then of course some specialized books, that are corner stones in math, but not especially easy to read, at least for beginners. But Andre Weil is a central figure in math. His younger sister Simone Weil was an author and philosopher, and a political activist on the left in French politics in the 1930ties. She died young.

Quite some autocensorship
Weil did indeed lead a colorful life, and he certainly was a dominant figure in maths. But if one writes an autobiography, he should be honest when speaking both of his best and his worst. Unlike this, Weil strictlyrestrains from the reader the chapters of his life which he is not so proudof. I will name just two. First, in the book his wife (with a young sonfrom her previous marriage) simply occurs in his life, and the reader isnever told how and where they met, and who was the husband she abandonedbecause of this acquaintance (I here only comment that it was anotherFrench mathematician of the period, and earlier a friend of Weil, but he isnot mentioned anywhere in the story). And second, his evading the militaryservice; surely, most of us do not want to die, but to explain that we donot want to serve because we have a hinduistic view of the World is anoverused trick (though perhaps it was not overused in those times). One sodevoted to that view would definitely not feel at home in Princeton. Inaddition, in his well-known style Weil is extremely critical of theprofessional qualities of practically everyone, with exception of his"family" - the Bourbaki. Most of the textbooks by other peoplewere awful, the knowledge of most other people catastrophicallyincomplete.

Still, there are some interesting facts for those who want toknow what a life of a mathematician can involve, and two stars are justlyearned. Perhaps this would have been a better book has it been written morein the spirit of Hardy's Apology, omitting the family matters and focusingon what the title promises.

Excellent!
Weil led a colorful, fascinating life.He became a well-known mathematician at an early age, and mastered several languages.Additionally, he traveled around the world, and narrowly escaped execution. His autobiography allows the reader to come to know him for himself,rather than for his work.If the reader pays close attention, he/she willsee his subtle humor, which is cleverly woven in to his life story. ... Read more

Book Description
This digital document, covering the life and work of Andre Weil, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 2102 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
• Education
• Professional associations and honors
• Employment
• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author

Book Description

"As a contribution to the history of mathematics, this is a model of its kind. While adhering to the basic outlook of Eisenstein and Kronecker, it provides new insight into their work in the light of subsequent developments, right up to the present day. As one would expect from this author, it also contains some pertinent comments looking into the future. It is not however just a chapter in the history of our subject, but a wide-ranging survey of one of the most active branches of mathematics at the present time. The book has its own very individual flavour, reflecting a sort of combined Eisenstein-Kronecker-Weil personality. Based essentially on Eisenstein's approach to elliptic functions via infinite series over lattices in the complex plane, it stretches back to the very beginnings on the one hand and reaches forward to some of the most recent research work on the other. (...) The persistent reader will be richly rewarded."
A. Fröhlich, Bulletin of the London Mathematical Society, 1978

Book Description
This volume contains the work of the great Swiss mathematician on differential geometry, a field marked by some of his greatest achievements. Between 1690 and 1700, Jacob Bernoulli published twelve treatises in the scientific journal Acta Eruditorum on the use of infinitesimal methods to answer geometrical questions. Preparatory notes for most of these papers and on many other themes are found in Bernoulli's scientific diary Meditationes, from which twentynine texts are published here for the first time. Among the curves considered are the isochrones (lines of constant descent), the parabolic spiral, the loxodrome, the cycloid, the tractrix, and the logarithmic spiral (Bernoulli's spira mirabilis, which also adorns his tombstone). The description of these curves by differential equations and by geometrical constructions, their rectification and quadrature, and the determination of their evolutes and caustics offered Bernoulli and his colleagues a range of challenging problems, many of them relevant for mechanical or optical applications. The French mathematician André Weil, who lived in the United States until his recent death, has greatly influenced 20th century mathematics, among other things, as a founding member of the Bourbaki group. For many years he has pursued intensive studies of the history of mathematics, especially number theory and algebraic geometry. Weil's introduction to this volume places Jacob Bernoulli's contribution to differential geometry in a line of development from Descartes, Huygens and Barrow through Newton's und Leibniz's epochal innovations right up to the codification of the subject by Euler. Martin Mattmüller, secretary of the Bernoulli Edition at Basel, edited the source text. His commentaries consider particular topics in differential geometry with reference to their historical context at the end of the 17th century. ... Read more

Book Description
Jacob Bernoulli (1654-1705), founder of Basel's dynasty of mathematicians, left a relatively small body of correspondence for posterity. Published here for the first time are all 50 preserved letters sent to or by Jacob Bernoulli, and documentary evidence is presented of some l00 further letters no longer extant. The correspondence with Leibniz takes up the bulk of the volume and is also the most substantial scientifically. Topics range from Bernoulli's theories of elasticity and probability, to Leibniz' "Dynamik", to the integration of algebraic functions, to controversies over differential calculus and calculus of variations. The exchange offers rare insights into the personal histories and working habits of these two correspondents. Science politics dominates the brief but intensive correspondence (1700-01) with Geneva's Nicolas Fatio de Duillier, a leading exponent of Newtonian mathematics. Their letters broach Leibniz' claim to mathematical leadership, professional disputes between the brothers Jacob and Johann, and Fatio's theory of gravitation. Bernoulli's letters to teachers and friends, and business letters to Otto Mencke, publisher of Acta Eruditorum, shed further light on the until now little-researched life of Basel's famed mathematician. ... Read more

Book Description
This classic is one of the cornerstones of modern algebraicgeometry. At the same time, it is entirely self-contained,assuming no knowledge whatsoever of algebraic geometry, and noknowledge of modern algebra beyond the simplest facts aboutabstract fields and their extensions, and the bare rudiments ofthe theory of ideals. ... Read more

Book Description

Articles by twenty-six leading mathematicians, all reflecting the influence of one of the seminal figures in modern mathematics, Andre Weil.

Customer Reviews (1)

Wonderful reading, will make you smile, if not laugh.
This is a monograph (in English, even though the title is in French) of two lectures given by Andre Weil in the early 60's showing the continuity of Number theory from Euler to Riemmann and beyond. Patrick Gallagher recommended it to me, and I am grateful to him for this.

Weil's lectures are full of witticisms, that perhaps only he could pull off. e.g. Weil claims that Analytic Number Theory is not Number Theory, and says that Hadamard was certainly not a Number Theorist (he knew him well). He also shows Euler's version of the functional equation of the Zeta function and guides the reader through a wonderful historical panorama of people and results related to the Zeta Function. ... Read more