Alexander Grothendieck

Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. He was 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.

Mathematical achievements

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.

Related Topics:
Sheaf - Jean-Pierre Serre - Kiyoshi Oka - Jean Leray

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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.

Related Topics:
Morphism - Riemann-Roch theorem - 1956 - Hirzebruch - Grothendieck-Riemann-Roch theorem - Arbeitstagung - Bonn - Armand Borel

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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 major field, because of its great expressive power 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.

Related Topics:
Algebraic geometry - Schemes - Nilpotent - Birational geometry - Number theory - Galois theory - Commutative algebra - Algebraic topology

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Its influence spilled over into many other branches of mathematics, for example the contemporary theory of D-modules. (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.)

Related Topics:
D-module - Maître à penser - Maître-penseur

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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). 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 largely withdrawn from mathematics.

Related Topics:
Éléments de géométrie algébrique - Séminaire de géométrie algébrique - étale - André Weil - Finite field - Complex number - Weil conjecture - Pierre Deligne

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Major mathematical topics (from Récoltes et Semailles)

He wrote a retrospective assessment of his mathematical work (see the external link La Vision below). As his main mathematical achievements ("maître-thèmes"), he chose this collection of 12 topics (his chronological order):

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• Topological tensor products and nuclear spaces
• "Continual" and "discrete" duality (derived categories and "six operations").
• Yoga of the Grothendieck-Riemann-Roch theorem (K-theory, relation with intersection theory).
• Schemes.
• Toposes.
• Étale cohomology including l-adic cohomology.
• Motives and the motivic Galois group (and Grothendieck categories)
• Crystals and crystalline cohomology, yoga of De Rham and Hodge coefficients.
• Topological algebra, infinity-stacks, 'dérivateurs', cohomological formalism of toposes as an inspiration for a new homotopic algebra
• Tame topology.
• Yoga of anabelian geometry and Galois-Teichmüller theory.
• Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.

He wrote that the central theme of the above is that of topos theory, while the first and last were of the least importance to him.

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Here the usage of yoga means a kind of 'meta-theory' that can be used heuristically.

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