Alexandre Grothendieck: A Mathematical Portrait (English and French Edition) by [various contributors] (PDF)

Description

Mathematics students and researchers often react to Alexandre Grothendieck’s legendary fame in the world of mathematics by asking just what the man did to earn him so brilliant a reputation. But as legitimate as it is, the question is difficult to answer, because of the particularly abstruse nature of his mathematics and the wealth of notions he introduced, some of which have become so natural and so familiar to mathematicians that it is easy to forget that there was ever an actual individual who first brought them out into the light, whereas others are so abstract that even experts may take years to grasp them. It is not merely a matter of stating some powerful, striking result that he proved, although there are many of these. But more deeply, Grothendieck’s work is based on a whole system of recasting old ideas in new ways, and it transformed the entire area of algebraic geometry essentially beyond recognition. This book attempts to provide a reasonable explanation of what made Grothendieck the mathematician that he was. Thirteen articles written by people who knew him personally — some who even studied or collaborated with him over a period of many years — portray Grothendieck at work, explaining the nature of his thought through descriptions of his discoveries and contributions to various subjects, and with impressions, memories, anecdotes, and some biographical elements. Seeing him through the eyes of those who knew him well, the reader will come away with a better understanding of what made Grothendieck unique. This volume also features 6 pages of photographs of the contributing authors.

User’s Reviews

Editorial Reviews: Review … [The contributors] write with great insight on Grothendieck’s mathematical legacy. What comes over strongly is his immense productivity during the early decades of his career, often relying on students and colleagues to write up his prodigious output, and his desire for mathematics to be as natural and as general as possible, often delaying publication until the exposition fitted perfectly into his global view of mathematics. This collage of viewpoints provides a fascinating cubist portrait of Grothendieck, which will form an excellent complement to the multi-part biography which Wilfried Scharlau is writing… –Gareth Jones, London Mathematical Society Newsletter, May 2015The present book is a compendium or a collage of articles having to do with the different facets of Grothendieck both as a hugely important and influential scholar and as an ultimately enigmatic individual with a remarkable history, including a past filled with childhood tragedy and strife, and a present full of mystery. There are articles by mathematicians, experts all, on the major themes of Grothendieck as an unsurpassed algebraic geometer. The book under review is irresistible to anyone who has even a mild interest in and acquaintance with algebraic geometry … and who is fascinated by Grothendieck’s remarkable life, including a trajectory from a fatherless childhood as a refugee from the Nazis sheltered in southern France, to early adulthood as an outsider who nonetheless absorbed mathematics through his pores, to a stellar career as one of the unquestioned grand masters of algebraic geometry, doing things in magical and radical ways in a style all his own, and then to the mystery that he now poses. The authors represented in this “Mathematical Portrait” are uniquely positioned to comment not only on Grothendieck’s mathematics but on the man himself, his personality, his influence (and his influence on them), and his uniqueness. Yes indeed, I think the book is altogether irresistible. –Prof. Michael Berg, Loyola Marymount University, Los Angeles, Calif. (for MAA Reviews)The remarkable life of Grothendieck has shaped not only the work but also the lives of the authors in this book and many more. What the readers will find in this fascinating collection of articles is not only valuable overviews of aspects of Grothendieck’s work but vivid glimpses of what it was like to be drawn into Grothendieck’s orbit. –Jeremy Gray, Metascience, Nov. 2014, Springer. About the Author Leila Schneps studied mathematics at Harvard University and then went on to do her doctoral thesis at the University of Paris, where she is now a researcher. She encountered Grothendieck’s unpublished writings in 1991, and her mathematical work for over two decades has been devoted to furthering some of the ideas sketched in them, and relating them to other areas of mathematics.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Alexandre Grothendieck: A Mathematical PortraitEdited by Leila SchnepsAlexandre Grothendieck (1928 – 2014) was the leading figure in the creation of modern algebraic geometry extending over many fields leading to revolutionary advances in many areas of pure mathematics. Leading a ‘Golden Age’ of mathematics 1959 – 70 at the Institut des Hautes Études Scientifiques (IHES) just south of Paris he walked away from his job isolating himself from his mathematical companions at the age of 42. How does a layman, with some mathematical background, get an appreciation of such a mathematician and his mathematics? This book is a real opportunity without any “dumbing down”.To quote from the Forward:”This book aims to explain, to generations of young people who didn’t even begin to study mathematics until long after Alexandre Grothendieck totally disappeared from the scene, the special and extraordinary nature of his contribution to mathematics.””Grothendieck’s identifying traits were the search for maximum generality, the focus on the harmonious aspect of structure, the lack of interest in special cases, the transfer of attention from objects themselves to morphisms between them, and – perhaps most appealingly – Grothendieck’s unique approach to difficulties that consisted in turning them, somehow, upside down, and making them into the actual central point and object of study, an attitude which has the power to subtly change them from annoying obstacles into valuable tools that actually help solve problems and prove theorems.””It was the absolute simplicity (in his own words “nobody before me has dipped so low”) and the total freshness and fearlessness of his vision, seemingly unaffected by long-established views and vantage points, that made Grothendieck who he was.””Like a cubist work of art, the mathematical portrait of Grothendieck painted in this book is made up of a multiplicity of different planes and different angles.”To achieve this, the book consists of 13 articles of commentary and anecdote without sparing any of the mathematics. The general reader can share the portrait and absorb the atmosphere from enjoyable bits and pieces directly from those who knew Grothendieck personally, although unfortunately three of the articles are in French that (to a non-French-speaking person) neatly renders the commentary as well as the mathematics impossible.Another quote “A scientific biography should be written in which we indicate the “flow” of mathematics … discussing a certain aspect of Grothendieck’s work, indicating possible routes, then describing the leap Grothendieck made from those roots to general ideas, and finally setting forth the impact of those ideas”.Grothendieck was a coordinator who got others interested in topics with a kind of amazing insight into what was a good problem to give a particular person to think about. And he was somehow mathematically incredibly charismatic, so that it seemed like people felt it was almost a privilege to be asked to do something that was part of Grothendieck’s long-range vision of the future. He had an instinct for “matching people with open problems. He would size you up and pose a problem that would be just the thing to illuminate the world for you. It’s a mode of perceptiveness that it’s quite wonderful and rare”. “Grothendieck certainly did not feel that he was attempting to use powerful techniques in order to obtain stronger results by generalising. What he perceived himself as doing was simplifying situations and objects by extracting the fundamental essence of the structure”.Obtaining even good results “the wrong way” – using clever tricks to get around deep theoretical obstacles – could infuriate Grothendeick. And his mathematics was pure so as to be compatible with his “anti-war” advocacy.”A certain breeze has died away” now … little remains of Grothendieck’s school. This is the reason to recommend this book perhaps understanding less than 5% of the content. It is an opportunity to feel the mathematical atmosphere of that time and appreciate this mathematician.Malcolm Cameron12 September 2015

⭐Der in Berlin geborene Alexander Grothendieck (1928-2014) gilt für viele als der größte Mathematiker des 20. Jahrhunderts. Dennoch ist sein Name in der allgemeinen Bevölkerung gänzlich unbekannt, auch etwa Mathematikstudenten begegnen in ihrem Studium seinem Namen eher selten. Grothendieck hat ein außerordentlich bewegtes Leben gehabt als Sohn anarchistisch-aktiver Eltern. Sein Vater russisch-jüdischer Herkunft wurde in Auschwitz ermordet. Grothendieck konnte mit seiner Mutter in Frankreich überleben. Grothendiecks aktives mathematisches Wirken dauerte nur etwa 20 Jahre. Umso beeindruckender ist seine Leistung. Er schuf für die Algebraische Geometrie mit einer genialen Weitsicht, Geschwindigkeit und Abstraktionsfähigkeit eine völlig neue Grundlage und Methodik, ohne die spätere Beweise einiger berühmter Vermutungen wohl nicht möglich gewesen wäre: Die Weilschen Vermutungen (Analoga der Riemannschen Vermutung über die Zeta-Funktion), bewiesen durch Grothendiecks Schüler Deligne, die Mordellsche Vermutung, bewiesen durch Faltings, und die 350 Jahre alte Fermatsche Vermutung, bewiesen durch Wiles. Beeinflusst duch die 1968er Bewegung zog sich Grothendieck auf dem Höhepunkt seiner Karriere mit nur 42 Jahren — unerklärlich für die mathematische Welt — völlig von der aktiven Forschung zurück, arbeitete ein paar Jahre in der Friedensbewegung. Die letzten 20 Jahre seines Lebens hat er in völliger Isolation in einem kleinen Ort in den Pyrenäen verbracht. Grothendieck hinterlässt viele tausend Manuskriptseiten, auch viele mathematische, die noch auf ihre Auswertung warten. Vieles im Leben und Schaffen von Grothendieck ist rätselhaft.Das vorliegende Buch (kurz vor Grothendiecks Tod veröffentlich) versucht sein mathematisches Werk zu würdigen. Es enthält 13 Artikel verschiedener Autoren (10 in englisch, 3 in französisch), deren Autoren allesamt Grothendieck persönlich kannten, z. T. mit ihm zusammengearbeitet haben und selbst zur Crème de la Crème in den jeweiligen Gebieten zählen. So werden einige wesentliche Errungenschaften Grothendiecks eher überblicksartig (aber nicht oberflächlich) beschrieben: etwa topologische Tensorprodukte und nukleare Räume, Schemata, K-Theorie, Descent (Grothendieck Topologien, Stacks…), Picard Schema, Fundamentalgruppe, Étale Kohomologie, Topoi, Motive… Auch wird ein Teil der umfangreichen Briefkorrespondenz Grothendiecks mit Jean-Pierre Serre dargestellt. Viele Artikel sind gespickt mit persönlichen Bemerkungen zur Person und Begegnung. Besonders Pierre Cartier gibt auch einige biografische Details und Erklärungsversuche für den Bruchseines Kollegen an. David Mumford erklärt kurz und prägnant (beispielhaft) den Stand der damaligen Algebraische Geometrie als Grothendieck die Bühne betrat und worin dessen Neuerung bestand. Robin Hartshorne berichtet u. a. über die Entstehung seiner Lecture Notes “Residue and Duality”. (Das komplette Inhaltsverzeichnis findet man in der Vorschau.)Jedem, der an Grothendiecks mathematischem Werk interessiert ist, kann ich dieses Buch (trotz des hohen Preises) empfehlen. Es gibt m. E. einen sehr guten Überblick und sehr schöne und kompetente Einführungen in die jeweiligen Gebiete (etwa Carlos Simpsons Artikel über “Descent”). Es ist sehr angenehm, diese schöne Sammlung in einem Band zu haben. Es ist natürlich zu bedenken, wie der Untertitel “A mathematical Portrait” ja nahelegt, dass die meisten Texte ein recht hohes Maß an mathematischem Verständnis erfordern, auch wenn die Artikel überblicksartig bleiben.

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