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Because of his mastery of abstract approaches to mathematics, but also because of the many stories told about his retirement and his alleged mental disorders, he is one of the most intriguing scientific personalities of the 20th century.
Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had been invented by Kiyoshi OkaKiyoshi Oka ( , April 19, 1901 March 1, 1978) was a Japanese mathematician, who did fundamental work in the theory of several complex variables. He was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduat and Jean LerayJean Leray ( 7 November 1906- 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. He was born in Nantes. His main work in topology was carried out while he was in a prisoner of war camp i. 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 morphismIn mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. In set theory, for example, morphisms are just functions, in group theory they are gro), allowing a broad generalization of many classical theorems. This he applied first to the Riemann-Roch theoremIn mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the compl, around 19561956 is a leap year starting on Sunday. see link for calendar) Events January January 1 End of Anglo- Egyptian Condominium in Sudan. January 16 President Gamal Abdal Nasser of Egypt vows to reconquer Palestine January 26 1956 Winter Olympic Games open in, which had already recently been generalized to any dimension by HirzebruchFriedrich E. Hirzebruch (born 17 October 1927) is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He was born in Hamm, Westphalia. He studied at the University of). The Grothendieck-Riemann-Roch theorem was announced by Grothendieck at the initial Arbeitstagung in BonnBonn is a city in Germany (Population (2002 est): 310 930), in the Bundesland of North Rhine-Westphalia, located ca. 20 kilometres south of Cologne on the river Rhine. It was the capital of West Germany from 1949 to 1990. The history of the city dates bac, in 1957. It appeared in print in a paper written by Armand Borel with Serre.
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 work is at a higher level of abstraction than prior versions of algebraic geometry. His theory of schemes has become established as the best universal foundation for this major field, due to its great power. 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.
Its influence spilled over into many other branches of mathematics, for example the contemporary theory of D-module s. (It also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems. Grothendieck is one of the few mathematicians who matches the French concept of maître à penser ; some go further and call him maître-penseur .)
The bulk of Grothendieck's published work is collected in the monumental, and yet incomplete, Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA). This is considered to have gone a long way to answering Kronecker's wish for a foundational theory based on the integers alone, by using the relative point of view based on finitely-generated commutative rings. The style of the mathematics is very distant from Kronecker's, though. On the EGA project Grothendieck collaborated with Jean Dieudonné. It is axiomatic, and claims descent (according to Dieudonné) from David Hilbert's approach; as interpreted by Nicolas Bourbaki. On the other hand Grothendieck himself applied an intuitive approach, as well as a generalising one. There were many other contributors to the SGA series.
Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers. Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck. This program culminated in the proofs of the Weil conjectures by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had withdrawn from mathematics.