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Who is Alexander Grothendieck? Anarchy, mathematics, spirituality. A biography. Part 1: Anarchy. (Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität. Eine Biographie. Teil 1: Anarchie.) (German) Zbl 1129.01018
Havixbeck: Scharlau. 180 p. (2007).
Alexander Grothendieck, born March 28, 1928 in Berlin, Germany, is a mathematician of French citizenship, who unanimously is considered to be one of the greatest minds in contemporary mathematics as a whole. His influence on the development of several branches of modern pure mathematics in the second half of the 20th century was absolutely pioneering and propelling, especially with regard to algebraic geometry, homological algebra, topology, arithmetic geometry, and functional analysis. He was awarded the Fields Medal in 1966 for his utmost fundamental contributions to the rigorous, abstract and sweeping reformation of algebraic geometry by developing completely new conceptual frameworks, methods, and tools of most powerful impact. In particular, Alexander Grothendieck became famous for his great mastery of highly abstract approaches to classical mathematics, for his unparalleled perfectionism in matters of systematizing and presenting theories, and for demonstrating how to derive new concrete results from very general methods.
On the other hand, apart from the unique mathematical genius Alexander Grothendieck, there is the just as unique personality of this outstanding man as a contemporary, socially and politically acting human being, colleague, teacher, next of kin, friend, and adversary, which is the subject of many haunting stories and some misleading rumors about him. The book under review, written by a leading German mathematician and established novel writer, is the first part of a biographical trilogy about the uniquely fascinating and dramatic life of the “phenomenon” Alexander Grothendieck. Based upon extensive investigations, consultations of acquaintances of Grothendieck’s, interpretations of some of his correspondences, and analyzing his 1000-page autobiographical manuscript “Récoltes et semailles” (1986), W. Scharlau has undertaken the rewarding attempt to present an authentic, objective and fair report on the first third of Alexander Grothendieck’s exceptional life, thereby trying to illuminate the essential facts that formed his individual characteristic, philosophy of life, and social behaviour later on.
Accordingly, the current first volume is concerned with his ancestry, his parents, his early family life, his fate as a displaced person during the upheavals of World War II, his school time, his studies in mathematics, and his early career as a mathematical researcher in France.
The first nine sections of the book are devoted to the life story of Alexander Grothendieck’s Russian Jewish father, Alexander Shapiro (1890–1942), and his anarchist political activities which finally ended with his tragic death in Auschwitz. The following four sections give a first depiction of Alexander Grothendieck’s German mother, Johanna (Hanka) Grothendieck (1900–1957), her parents, her youth, and her first husband, Johannes (Alf) Raddatz (1897–1958). Born as Alexander Raddatz in 1928 in Berlin, Alexander Grothendieck spent the first five years of his life, together with his mother and his half-sister, Frode (Maidi) Raddatz, in Berlin, before having been entrusted to his foster-parents, the Heydorn family in Hamburg-Blankenese, when his mother Hanka joint his father, Alexander Shapiro (alias Tanaroff) in their political activities in France and Spain. This is described in Sections 14–20, together with the known facts about the end of Grothendieck’s father in Auschwitz, while the following sections compile various details about the joint life of Hanka and Alexander Grothendieck in diverse refugee camps in France between 1940 and 1944 (Sections 21–23).
The remaining sections give particulars of the complicated relations between Alexander Grothendieck and his mother after 1945, of Alexander’s growing into a mathematician during his studies in Montpellier, and of Hanka Grothendieck’s unpublished autobiographical novel “A Woman” (“Eine Frau”, written in German).
The present first volume concludes the report on Alexander Grothendieck’s first 25 years of life with a description of his extended mathematical studies in Paris and Nancy, his rise to a great researcher under the influence of L. Schwartz, H. Cartan, J. Dieudonné, J.-P. Serre, and others, and of the dramatic clashes between him and his mother Hanka until her death in 1957.
Actually, the biographies of his parents, foster-parents, siblings, and other related folks play a predominant role in this part of the trilogy, just as the various unusual circumstances in his early life do, but it is exactly these facts that help understand the shaping of Alexander Grothendieck’s unique individuality during that crucial period of his growing up. The author portrays Grothendieck’s early associates and their paths through life in a very detailed and gripping manner, and that with a masterly feeling for correct idiom, and he places the background of the young Alexander Grothendieck precisely into this web of almost bizarre marches of destiny. No doubt, this is an essential key to understand his later mode of life, his unusual habits and politics, his confrontations with colleagues and French authorities, his left-wing and pacifist political views, his early withdrawal from mathematics, his reasons for retirement, his later writings, and finally his disappearance from social life in 1991.
In the forthcoming two volumes, the author plans to tell about Alexander Grothendieck’s creative work as active researcher in mathematics during the period from 1950 until 1970, when he virtually revolutionized large parts of pure mathematics, and about his withdrawal from both mathematics and the human society afterwards, his meditations, and his mysterious behaviour ever since.
Winfried Scharlau is completely right when he emphasizes that the nearly incredible path of life of Alexander Grothendieck should be regarded as a vivid part of our society in the 20th century and thereafter, thus deserving very much interest, analysis, and profound understanding. He has portrayed the first third of Grothendieck’s life without the slightest trace of sensationalism, as it shows through in many rumors about it, but with a sympathetic understanding for an exceptional human being whose spirituality and moral philosophy seems to have reached the borderline of human nature.

01A70 Biographies, obituaries, personalia, bibliographies
01A60 History of mathematics in the 20th century
01A72 Schools of mathematics
14-03 History of algebraic geometry
01A65 Development of contemporary mathematics
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