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Would you also like to submit a review for this item? You already recently rated this item. Your rating has been recorded. Write a review Rate this item: 1 2 3 4 5. Preview this item Preview this item. Who is Alexander Grothendieck? After the war, the young Grothendieck studied mathematics in France, initially at the University of Montpellier where he did not initially perform well, failing such classes as astronomy. After three years of increasingly independent studies there, he went to continue his studies in Paris in By , he set this subject aside in order to work in algebraic geometry and homological algebra.
The prospect did not worry him, as long as he could have access to books. He was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom is all the more incredible; quite unique in the history of mathematics.
He was, however, able to play a dominant role in mathematics for around a decade, gathering a strong school. Jean Giraud worked out torsor theory extensions of nonabelian cohomology. Many others were involved.
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Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology relevant also in categorical logic. He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory. As a framework for his coherent duality theory he also introduced derived categories , which were further developed by Verdier. Grothendieck's political views were radical and pacifist , and he strongly opposed both United States intervention in Vietnam and Soviet military expansionism.
He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam War. The Grothendieck Festschrift , published in , was a three-volume collection of research papers to mark his sixtieth birthday in In it, Cartier notes that as the son of an antimilitary anarchist and one who grew up among the disenfranchised, Grothendieck always had a deep compassion for the poor and the downtrodden.
By the late s, he had started to become interested in scientific areas outside mathematics. In , Grothendieck, with two other mathematicians, Claude Chevalley and Pierre Samuel , created a political group called Survivre —the name later changed to Survivre et vivre. The group published a bulletin and was dedicated to antimilitary and ecological issues, and also developed strong criticism of the indiscriminate use of science and technology.
While not publishing mathematical research in conventional ways during the s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content. In , stimulated by correspondence with Ronald Brown and Tim Porter at Bangor University , Grothendieck wrote a page manuscript titled Pursuing Stacks , starting with a letter addressed to Daniel Quillen.
This letter and successive parts were distributed from Bangor see External links below. Within these, in an informal, diary-like manner, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks. The manuscript, which is being edited for publication by G. Written in , this latter opus of about pages further developed the homotopical ideas begun in Pursuing Stacks. It describes new ideas for studying the moduli space of complex curves. Although Grothendieck himself never published his work in this area, the proposal inspired other mathematicians' work by becoming the source of dessin d'enfant theory and Anabelian geometry.
He complains about what he saw as the "burial" of his work and betrayal by his former students and colleagues after he had left the community. In Grothendieck declined the Crafoord Prize with an open letter to the media. He wrote that established mathematicians like himself had no need for additional financial support and criticized what he saw as the declining ethics of the scientific community, characterized by outright scientific theft that, according to him, had become commonplace and tolerated. The letter also expressed his belief that totally unforeseen events before the end of the century would lead to an unprecedented collapse of civilization.
Grothendieck added however that his views are "in no way meant as a criticism of the Royal Academy's aims in the administration of its funds" and added "I regret the inconvenience that my refusal to accept the Crafoord prize may have caused you and the Royal Academy. La Clef des Songes , a page manuscript written in , is Grothendieck's account of how his consideration of the source of dreams led him to conclude that God exists. In it, he described his encounters with a deity and announced that a "New Age" would commence on 14 October Over 20, pages of Grothendieck's mathematical and other writings, held at the University of Montpellier, remain unpublished.
In , Grothendieck moved to a new address which he did not provide to his previous contacts in the mathematical community.
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Local villagers helped sustain him with a more varied diet after he tried to live on a staple of dandelion soup. He asks that none of his work be reproduced in whole or in part and that copies of this work be removed from libraries. Grothendieck was born in Weimar Germany. In , aged ten, he moved to France as a refugee. Records of his nationality were destroyed in the fall of Germany in and he did not apply for French citizenship after the war.
He thus became a stateless person for at least the majority of his working life, traveling on a Nansen passport.
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Grothendieck was very close to his mother to whom he dedicated his dissertation. She died in from the tuberculosis that she contracted in camps for displaced persons. Grothendieck's early mathematical work was in functional analysis. His key contributions include topological tensor products of topological vector spaces , the theory of nuclear spaces as foundational for Schwartz distributions , and the application of L p spaces in studying linear maps between topological vector spaces.
It is, however, in algebraic geometry and related fields where Grothendieck did his most important and influential work. Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray. Grothendieck took them to a higher level of abstraction and turned them into a key organising principle of his theory. He shifted attention from the study of individual varieties to the relative point of view pairs of varieties related by a morphism , allowing a broad generalization of many classical theorems.
In , he applied the same thinking to the Riemann—Roch theorem , which had already recently been generalized to any dimension by Hirzebruch. This result was his first work in algebraic geometry.
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He went on to plan and execute a programme for rebuilding the foundations of algebraic geometry, which were then in a state of flux and under discussion in Claude Chevalley 's seminar; he outlined his programme in his talk at the International Congress of Mathematicians. His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions. He adapted the use of non-closed generic points , which led to the theory of schemes. He also pioneered the systematic use of nilpotents. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings.
His theory of schemes has become established as the best universal foundation for this field, because of its expressiveness as well as technical depth. In that setting one can use birational geometry , techniques from number theory , Galois theory and commutative algebra , and close analogues of the methods of algebraic topology , all in an integrated way.
He is also noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation. Part 1: Anarchy. Format Paperback. Publisher Books on Demand. Edition nd. See details. See all 2 brand new listings. Buy It Now. Add to cart. Part : Anarchy by Winfried Scharlau , Paperback.
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