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Encyclopedia > Alexander Grothendieck
Alexander Grothendieck
Image:Grothendieck.jpg
Born March 28, 1928
Berlin, Germany
Field mathematician
Known for algebraic geometry, homological algebra, and functional analysis
Notable prizes Fields Medal (1966)

Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) 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 which are still astounding in their broad scope and sheer bulk, and a lifestyle later in his career which alienated 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 in a letter to the media. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... March 28 is the 87th day of the year (88th in leap years) in the Gregorian calendar. ... Year 1928 (MCMXXVIII) was a leap year starting on Sunday (link will display full calendar). ... Location of Berlin within Germany / EU Coordinates Time zone CET/CEST (UTC+1/+2) Administration Country NUTS Region DE3 City subdivisions 12 boroughs Governing Mayor Klaus Wowereit (SPD) Governing parties SPD / Left. ... Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... The Crafoord Prize was established in 1980 by Holger Crafoord, the inventor of the artificial kidney, and his wife Anna-Greta Crafoord. ... Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ...


He is noted for his mastery of abstract approaches to mathematics, and his perfectionism in matters of formulation and presentation. In particular, his ability to derive concrete results using only very general methods is considered to be unique amongst mathematicians. 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. In academic publishing, a scientific journal is a periodical publication intended to further the progress of science, usually by reporting new research. ... Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... Harvard University (incorporated as The President and Fellows of Harvard College) is a private university in Cambridge, Massachusetts, USA and a member of the Ivy League. ...

Contents

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. In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Kiyoshi Oka (岡 潔, April 19, 1901 – March 1, 1978) was a Japanese mathematician, who did fundamental work in the theory of several complex variables. ... Jean Leray (7 November 1906-10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. ...


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 Mathematische Arbeitstagung in Bonn, in 1957. It appeared in print in a paper written by Armand Borel with Serre. Grothendiecks relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of objects explicitly depending on parameters, as the basic field of study, rather than a single such object. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In 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. ... Friedrich E.P. 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. ... In mathematics, specifically in algebraic geometry, the Grothendieck-Riemann-Roch theorem is a far-reaching result on coherent cohomology. ... The Mathematische Arbeitstagung taking place annually in Bonn since 1957, and founded by Friedrich Hirzebruch, was an international meeting of mathematicians intended to act in clearing-house fashion, by disseminating current research ideas; and, at the same time, to bring mathematics in West Germany back into its place in European... Bonn is the 19th largest city in Germany, located about 20 kilometres south of Cologne on the river Rhine in the Federal State of North Rhine-Westphalia. ... Armand Borel (21 May 1923 - 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study, Princeton from 1957 to 1993. ...


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. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, in the fields of general topology and particularly of algebraic geometry, a generic point P of a topological space X is a point such that every point Q of X is a specialization of P, in the sense of the specialization order (or pre-order). ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...


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.) In mathematics, a D-module is a module over a ring D of differential operators. ... Maître à penser is a French language phrase, denoting a teacher whom one chooses, in order to learn not just a set of facts or point of view, but a way of thinking. ... Maître à penser is a French language phrase, denoting a teacher whom one chooses, in order to learn not just a set of facts or point of view, but a way of thinking. ...


EGA and SGA

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). The collection Fondements de la Géometrie Algébrique (FGA), which gathers together talks given in the Séminaire Bourbaki, also contains important material. The Éléments de géométrie algébrique (Elements of Algebraic Geometry) by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, are an unfinished 1500-page treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut... In mathematics, Alexander Grothendiecks Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois... FGA, or Fondements de la Géometrie Algébrique, is a book that collected together seminar notes of Alexander Grothendieck. ... The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. ...


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. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...


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. The 'centre of gravity' of the SGA developments lies somewhat obliquely to the tools in fact required. In mathematics, the Weil conjectures, which had become theorems by 1974, were some highly-influential proposals from the late 1940s by André Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ...


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):

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

He wrote that the central theme of the topics above is that of topos theory, while the first and last were of the least importance to him. In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. ... In mathematics, a nuclear space is a topological vector space with many of the good properties of finite dimensional vector spaces. ... The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ... In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ... In mathematics, specifically in algebraic geometry, the Grothendieck-Riemann-Roch theorem is a far-reaching result on coherent cohomology. ... In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ... In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a topos (plural topoi or toposes) is a type of category that behaves like the category of sheaves of sets on a topological space. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... Crystal in homological algebra is a diagram with points and arrows. ... In mathematics, crystalline cohomology is a Weil cohomology theory for schemes discovered by Grothendieck (in his letter to Tate (Grothendieck 1966) and his lecture (Grothendieck 1968)) and developed by Pierre Berthelot (1974). ... Wikipedia does not yet have an article with this exact name. ... In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra as well as possibly the abelian aspects as special cases. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... In mathematics, a topos (plural topoi or toposes) is a type of category that behaves like the category of sheaves of sets on a topological space. ...


Here the usage of yoga means a kind of 'meta-theory' which can be used heuristically. The word yoke, meaning a linkage, is derived from the same Indo-European root.


Life

Family and early life

Born to a Russian Jewish father, Alexander Shapiro, and a German mother[1], Hanka Grothendieck, in Berlin. He was a displaced person during much of his childhood due to the upheavals of World War II. For other uses, see Jew (disambiguation). ... Location of Berlin within Germany / EU Coordinates Time zone CET/CEST (UTC+1/+2) Administration Country NUTS Region DE3 City subdivisions 12 boroughs Governing Mayor Klaus Wowereit (SPD) Governing parties SPD / Left. ... It has been suggested that this article or section be merged with forced migration. ... Combatants Allied powers: China France Great Britain Soviet Union United States and others Axis powers: Germany Italy Japan and others Commanders Chiang Kai-shek Charles de Gaulle Winston Churchill Joseph Stalin Franklin Roosevelt Adolf Hitler Benito Mussolini Hideki Tōjō Casualties Military dead: 17,000,000 Civilian dead: 33,000...


Alexander lived with his parents both of whom were anarchists, until 1933, in Berlin. At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Alexander with a family in Hamburg where he went to school. During this time, his parents fought in the Spanish Civil War. Anarchism is a generic term describing various political philosophies and social movements that advocate the elimination of hierarchy and imposed authority. ... Location of Berlin within Germany / EU Coordinates Time zone CET/CEST (UTC+1/+2) Administration Country NUTS Region DE3 City subdivisions 12 boroughs Governing Mayor Klaus Wowereit (SPD) Governing parties SPD / Left. ... City flag City coat of arms Motto: Fluctuat nec mergitur (Latin: Tossed by the waves, she does not sink) Paris Eiffel tower as seen from the esplanade du Trocadéro. ... Location Coordinates Time zone CET/CEST (UTC+1/+2) Administration Country NUTS Region DE6 First Mayor Ole von Beust (CDU) Governing party CDU Votes in Bundesrat 3 (from 69) Basic statistics Area  755 km² (292 sq mi) Population 1,754,317 (11/2006)[1]  - Density 2,324 /km² (6,018... Combatants Spanish Republic With the support of: Soviet Union[1] Nationalist Spain With the support of: Italy Germany Commanders Manuel Azaña Francisco Largo Caballero Juan Negrín Francisco Franco Gonzalo Queipo de Llano Emilio Mola José Sanjurjo Casualties 500,000[2] The Spanish Civil War was a major conflict...


During WWII

In 1939 Alexander came to France and lived in various camps for displaced persons with his mother, first at the Camp de Rieucros, spending 1942-44 at Le Chambon-sur-Lignon. His father was sent via Drancy to Auschwitz where he died in 1942. Le Chambon-sur-Lignon is a town and commune in the Haute-Loire département in the Auvergne région of southern France. ... Drancy is a commune in the northeastern suburbs of Paris, France. ... Auschwitz (Konzentrationslager Auschwitz) was the largest of the Nazi German concentration camps. ...


Studies and contact with research mathematics

After the war, young Grothendieck studied mathematics in France, initially at the University of Montpellier. He had decided to become a math teacher because he had been told that mathematical research had been completed early in the 20th century and there were no more open problems.[2] However, his talent was noticed, and he was encouraged to go to Paris in 1948. The University of Montpellier, (Université de Montpellier), is a French university in Montpellier. ... City flag City coat of arms Motto: Fluctuat nec mergitur (Latin: Tossed by the waves, she does not sink) Paris Eiffel tower as seen from the esplanade du Trocadéro. ...


Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Superieure, but lacking the necessary background to follow the high-powered seminar, he moved to the University of Nancy where he wrote his dissertation under Laurent Schwartz in functional analysis, from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. By 1957, he set this subject aside in order to work in algebraic geometry and homological algebra. The quadrangle at the main ENS building on rue dUlm is known as the Cour aux Ernests – the Ernests being the goldfish in the pond. ... University of Nancy is a regional French university founded in 1572 and consists of three branches — The University of Nancy 1, The University of Nancy 2, and the INPL, National Polytechnic Institute of Lorraine. ... Laurent Schwartz (5 March 1915 – 4 July 2002 in Paris) was a French mathematician. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...


The IHES years

Installed at the Institut des Hautes Études Scientifiques (IHES), Grothendieck attracted attention, first by his spectacular Grothendieck-Riemann-Roch theorem, and then by an intense and highly productive activity of seminars (de facto working groups drafting into foundational work some of the ablest French and other mathematicians of the younger generation). Grothendieck himself practically ceased publication of papers through the conventional, learned journal route. He was, however, able to play a dominant role in mathematics for around a decade, gathering a strong school. IHÉS main building The Institut des Hautes Études Scientifiques (I.H.É.S.) is a French institute supporting advanced research in mathematics and theoretical physics. ... In mathematics, specifically in algebraic geometry, the Grothendieck-Riemann-Roch theorem is a far-reaching result on coherent cohomology. ... In academic publishing, a scientific journal is a periodical publication intended to further the progress of science, usually by reporting new research. ...


During this time he had officially as students Michel Demazure (who worked on SGA3, on group schemes), Luc Illusie (cotangent complex), Michel Raynaud, Jean-Louis Verdier (cofounder of the derived category theory) and Pierre Deligne. Collaborators on the SGA projects also included Mike Artin (étale cohomology) and Nick Katz (monodromy theory and Lefschetz pencils). Jean Giraud worked out torsor theory extensions of non-abelian cohomology. Many others were involved. In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. ... Michel Raynaud (June 16, 1938) is a French mathematician working in algebraic geometry. ... Jean-Louis Verdier (1935 – 1989) was a French mathematician who invented derived categories and Verdier duality. ... In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ... Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ... Michael Artin Michael Artin (born 1934) is an American mathematician and a professor at MIT, known for his contributions to algebraic geometry. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... Nick Katz (Nicholas M. Katz) is an American mathematician, working in the fields of algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ... In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, in order to analyse the algebraic topology of an algebraic variety V. A pencil here is a particular kind of linear system of divisors on V, namely a one-parameter family, parametrised by the projective... There is another famous Jean Giraud, comics author. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ...


Politics and retreat from scientific community

Grothendieck's political views were considered to be radical left wing and pacifist. He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam war. He retired from scientific life around 1970, after having discovered the partly military funding of IHES (see pp. xii and xiii of SGA1, Springer Lecture Notes 224). He returned to academia a few years later as a professor at the University of Montpellier, where he stayed until his retirement in 1988. His criticisms of the scientific community are also contained in a letter written in 1988, in which he states the reasons for his refusal of the Crafoord Prize. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Hanoi (Vietnamese: Hà Ná»™i, Hán Tá»±: 河内)  , estimated population 3,145,300(2005), is the capital of Vietnam. ... Combatants Republic of Vietnam United States Republic of Korea Thailand Australia New Zealand The Philippines National Front for the Liberation of South Vietnam Democratic Republic of Vietnam People’s Republic of China Democratic Peoples Republic of Korea Strength US 1,000,000 South Korea 300,000 Australia 48,000... IHÉS main building The Institut des Hautes Études Scientifiques (I.H.É.S.) is a French institute supporting advanced research in mathematics and theoretical physics. ... Montpellier (Occitan Montpelhièr) is a city in the south of France. ... The Crafoord Prize was established in 1980 by Holger Crafoord, the inventor of the artificial kidney, and his wife Anna-Greta Crafoord. ...


The Grothendieck Festschrift was a three-volume collection of research papers to mark his sixtieth birthday (falling in 1988), and published in 1990.[3]


Manuscripts written in the 1980s

While not publishing mathematical research in conventional ways during the 1980s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content. During that period he also released his work on Bertini type theorems contained in EGA 5, published by the Grothendieck Circle in 2004.


La Longue Marche à travers la théorie de Galois (roughly The Long Walk Through Galois Theory) is an approximately 1600-page handwritten manuscript produced by Grothendieck during the years 1980-1981, containing many of the ideas leading to the Esquisse d'un programme (see below) and in particular studying the Teichmüller theory.


In 1983 he wrote a huge extended manuscript (about 600 pages) entitled Pursuing Stacks, stimulated by correspondence with Ronnie Brown and Tim Porter at Bangor, and starting with a letter addressed to Daniel Quillen. This letter and successive parts were distributed from Bangor (see External Links below): in an informal manner, as a kind of diary, 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. Maltsiniotis, later led to another of his monumental works, Les Dérivateurs. Written in 1991, this latter opus of about 2000 pages further developed the homotopical ideas begun in Pursuing Stacks. Much of this work anticipated the subsequent development of the motivic homotopy theory of F. Morel and V. Voevodsky in the mid 1990s. Daniel Quillen (born June 21, 1940) is an American mathematician, a Fields Medallist, and the current Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In algebraic geometry, a branch of mathematics, an algebraic stack is a concept introduced to generalize algebraic varieties, schemes, and algebraic spaces. ... Vladimir Voevodsky (Russian: Владимир Воеводский) (born June 4, 1966) is a Russian mathematician. ...


His Esquisse d'un programme (1984) is a proposal for a position at the Centre National de la Recherche Scientifique, which he held from 1984 to his retirement in 1988. Ideas from it have proved influential, and have been developed by others, in particular in a new field emerging as anabelian geometry. In La Clef des Songes he explains how the reality of dreams convinced him of God's existence. The Centre national de la recherche scientifique (CNRS) is the largest and most prominent public research organization in France. ... A dream is the experience of envisioned images, sounds, or other sensations during sleep. ... This article discusses the term God in the context of monotheism and henotheism. ...


The 1000-page autobiographical manuscript Récoltes et semailles (1986) is now available on the internet in the French original, and an English translation is underway (these parts of Récoltes et semailles have already been translated into Russian and published in Moscow).


Disappearance

In 1991, he left his home and disappeared. He is now said to live in southern France or Andorra and to entertain no visitors. Though he has been inactive in mathematics for many years, he remains one of the greatest and most influential mathematicians of modern times.


See also

In mathematics, Grothendiecks Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. ... In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way. ... Grothendiecks relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of objects explicitly depending on parameters, as the basic field of study, rather than a single such object. ... In mathematics, specifically in algebraic geometry, the Grothendieck-Riemann-Roch theorem is a far-reaching result on coherent cohomology. ... In mathematics, Alexander Grothendiecks Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois... In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. ... In mathematics, a Grothendieck universe is a non-empty set U with the following properties: If x U and if y x, then y U. If x,y U, then {x,y} U. If x U, then P(x) U. (P(x) is the power set of x. ... In mathematics, the Grothendieck inequality relates to , where is the unit ball of a Hilbert space . ...

References

  1. ^ There are incompatibilities in the various accounts of his origins; see [1], [2] and [3]; Hanka's ethnic origin is uncertain.
  2. ^ See the first part of the two-part AMS article
  3. ^ The editors were Pierre Cartier, Luc Illusie, Nick Katz, Gérard Laumon, Yuri Manin, and Ken Ribet. A second edition has been printed (2007) by Birkhauser.

Pierre Cartier (born in Sedan, France in 1932) is a mathematician - more specifically, a category theorist. ... Nick Katz (Nicholas M. Katz) is an American mathematician, working in the fields of algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. ... Yuri Ivanovitch Manin (born 1937) is a Russian-born mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. ... Kenneth Alan Ken Ribet is an American mathematician, currently a professor of mathematics at the University of California, Berkeley. ...

External links

Persondata
NAME Grothendieck, Alexander
ALTERNATIVE NAMES
SHORT DESCRIPTION mathematician
DATE OF BIRTH March 28, 1928
PLACE OF BIRTH Berlin, Germany
DATE OF DEATH
PLACE OF DEATH

  Results from FactBites:
 
Alexander Grothendieck - Wikipedia, the free encyclopedia (1632 words)
Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century.
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.
Alexander lived with his father, Alexander Shapiro, and his mother, Hanka Grothendieck, both of whom were socialist revolutionaries.
Alexander Grothendieck (503 words)
Alexander Grothendieck, born in 1928 in Berlin, is one of the leading mathematicians of the twentieth century, with monumental contributions to functional analysis and algebraic geometry.
Perhaps Grothendieck's deepest accomplishment is the invention of the etale and l-adic cohomology theories, which explain an observation of Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.
Grothendieck's radical left-wing and pacifist politics were doubtless born of his wartime experiences.
  More results at FactBites »

 
 

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