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Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. He is also one of its most extreme scientific personalities, with achievements over a short span of years that are still scarcely credible in their broad scope and sheer bulk, and an approach that antagonised even close followers. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and co-awarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize, on ethical grounds.

He is noted for his mastery of abstract approaches to mathematics, and his perfectionism in matters of formulation and presentation. Relatively little of his work after 1960 was published by the conventional route of the learned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal, on French mathematics and the Zariski school at Harvard University. He is the subject of many stories and some misleading rumors, concerning his work habits and politics, confrontations with other mathematicians and the French authorities, his withdrawal from mathematics at age 42, his retirement and his subsequent lengthy writings.

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had been invented by Kiyoshi Oka and Jean Leray. Grothendieck took them to a higher level, changing the tools and the level of abstraction.

Amongst his insights, 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. This he applied first to the Riemann-Roch theorem, around 1956, which had already recently been generalized to any dimension by Hirzebruch. The Grothendieck-Riemann-Roch theorem was announced by Grothendieck at the initial Arbeitstagung in Bonn, in 1957. It appeared in print in a paper written by Armand Borel with Serre.

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

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