Alexander Grothendieck
   HOME

TheInfoList



Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

mathematician
who became the leading figure in the creation of modern
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
. His research extended the scope of the field and added elements of
commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
,
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
,
sheaf theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and
category theory Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...
to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
. He is considered by many to be the greatest mathematician of the 20th century. Born in Germany, Grothendieck was raised and lived primarily in France, and he and his family were persecuted by the
Nazi regime Nazi Germany, (lit. "National Socialist State"), ' (lit. "Nazi State") for short; also ' (lit. "National Socialist Germany") officially known as the German Reich from 1933 until 1943, and the Greater German Reich from 1943 to 1945, was t ...

Nazi regime
. For much of his working life, however, he was, in effect, stateless. As he consistently spelled his first name "Alexander" rather than "Alexandre" and his surname, taken from his mother, was the Dutch-like
Low German Low German or Low Saxon (in the language itself: , and other names; german: Plattdeutsch, ) is a West Germanic languages, West Germanic language variety spoken mainly in Northern Germany and the northeastern part of the Netherlands. It is also spo ...
"Grothendieck", he was sometimes mistakenly believed to be of Dutch origin. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received his Fields Medal in 1966 for advances in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
,
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
, and
K-theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. He later became professor at the
University of Montpellier The University of Montpellier (french: Université de Montpellier) is a French public research university A research university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of highe ...
and, while still producing relevant mathematical work, he withdrew from the mathematical community and devoted himself to political and religious pursuits (first Buddhism and later a more Christian vision). In 1991, he moved to the French village of Lasserre in the
Pyrenees The Pyrenees (; es, Pirineos ; french: Pyrénées ; ca, Pirineus ; eu, Pirinioak ; oc, Pirenèus ; an, Pirineus) is a mountain range straddling the border of and . It extends nearly from its union with the to on the coast. It reaches a ma ...

Pyrenees
, where he lived in seclusion, still working tirelessly on mathematics until his death in 2014.


Life


Family and childhood

Grothendieck was born in
Berlin Berlin (; ) is the and by both area and population. Its 3,769,495 inhabitants, as of 31 December 2019 makes it the , according to population within city limits. One of 's , Berlin is surrounded by the state of , and contiguous with , Brande ...

Berlin
to
anarchist Anarchism is a political philosophy Political philosophy is the philosophical study of government, addressing questions about the nature, scope, and legitimacy of public agents and institutions and the relationships between them. Its top ...

anarchist
parents. His father, Alexander "Sascha" Schapiro (also known as Alexander Tanaroff), had
Hasidic Jewish Hasidism, sometimes spelled Chassidism, and also known as Hasidic Judaism ( he, חסידות, Ḥăsīdut, ; originally, "piety"), is a subgroup of Haredi Judaism that arose as a spiritual revival movement in the territory of contemporary Wester ...
roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck, came from a
Protestant Protestantism is a form of that originated with the 16th-century , a movement against what its followers perceived to be in the . Protestants originating in the Reformation reject the Roman Catholic doctrine of , but disagree among themselves ...
family in
Hamburg Hamburg (, ; nds, label=Hamburg German, Low Saxon, Hamborg ), officially the Free and Hanseatic City of Hamburg (german: Freie und Hansestadt Hamburg; nds, label=Low Saxon, Friee un Hansestadt Hamborg),. is the List of cities in Germany by popul ...

Hamburg
and worked as a journalist. Both had broken away from their early backgrounds in their teens. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and his birth name was initially recorded as "Alexander Raddatz." The marriage was dissolved in 1929 and Schapiro/Tanaroff acknowledged his paternity, but never married Hanka. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to
Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,175,601 residents , in an area of more than . Since the 17th century, Paris ha ...

Paris
to evade
Nazism Nazism (), officially National Socialism (german: Nationalsozialismus; ), is the ideology and practices associated with Adolf Hitler and the Nazi Party (german: link=no, Nationalsozialistische Deutsche Arbeiterpartei, NSDAP, or National So ...

Nazism
, followed soon thereafter by his mother. They left Grothendieck in the care of Wilhelm Heydorn, a
Lutheran Lutheranism is one of the largest branches of Protestantism Protestantism is a form of Christianity Christianity is an Abrahamic religions, Abrahamic Monotheism, monotheistic religion based on the Life of Jesus in the New Testament, life ...
pastor A pastor (abbreviated as "Pr" or "Ptr" , or "Ps" ) is the leader of a Christian Christians () are people who follow or adhere to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus in Christianity, Jesus ...

pastor
and teacher in
Hamburg Hamburg (, ; nds, label=Hamburg German, Low Saxon, Hamborg ), officially the Free and Hanseatic City of Hamburg (german: Freie und Hansestadt Hamburg; nds, label=Low Saxon, Friee un Hansestadt Hamborg),. is the List of cities in Germany by popul ...

Hamburg
. During this time, his parents took part in the
Spanish Civil War The Spanish Civil War ( es, Guerra Civil Española)) or The Revolution ( es, La Revolución) among Nationalists, the Fourth Carlist War ( es, Cuarta Guerra Carlista) among Carlists Carlism ( eu, Karlismo; ca, Carlisme; ; ) is a Traditiona ...

Spanish Civil War
, according to
Winfried Scharlau Winfried Scharlau (born 12 August 1940 in Berlin, died 26 November 2020) was a German mathematician. Biography Scharlau received his doctorate in 1967 from the University of Bonn. His doctoral thesis ''Quadratische Formen und Galois-Cohomologie'' ( ...
, as non-combatant auxiliaries, though others state that Sascha fought in the anarchist militia.


World War II

In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterwards his father was interned in Camp Vernet, Le Vernet. He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners". The first was the Rieucros Camp, where his mother contracted the tuberculosis which eventually caused her death and where Alexander managed to attend the local school, at Mende, Lozère, Mende. Once Alexander managed to escape from the camp, intending to assassinate Hitler. Later, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II.Amir D. Acze
''The Artist and the Mathematician,''
Basic Books, 2009 pp.8ff.pp.8-15.
Alexander was permitted to live, separated from his mother,Luca Barbieri Viale, 'Alexander Grothendieck:entusiasmo e creatività,' in C. Bartocci, R. Betti, A. Guerraggio, R. Lucchetti (eds.,
''Vite matematiche: Protagonisti del '900, da Hilbert a Wiles,''
Springer Science & Business Media, 2007 pp.237-249 p.237.
in the village of Le Chambon-sur-Lignon, sheltered and hidden in local boarding houses or Pension (lodging), pensions, though he occasionally had to seek refuge in the woods during Nazis raids, surviving at times without food or water for several days. His father was arrested under the Vichy anti-Jewish legislation, and sent to the Drancy internment camp, Drancy, and then handed over by the Vichy France, French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Chambon, Grothendieck attended the Collège Cévenol (now known as the Le Collège-Lycée Cévenol International), a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Chambon attended Cévenol, and it was at this school that Grothendieck apparently first became fascinated with mathematics.


Studies and contact with research mathematics

After the war, the young Grothendieck studied mathematics in France, initially at the
University of Montpellier The University of Montpellier (french: Université de Montpellier) is a French public research university A research university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of highe ...
where he did not initially perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure. After three years of increasingly independent studies there, he went to continue his studies in Paris in 1948. Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil, he moved to the University of Nancy where two leading experts were working on Grothendieck's area of interest, Topological Vector Spaces: Jean Dieudonné and Laurent Schwartz. The latter had recently won a Fields Medal. He showed his new student his latest paper; it ended with a list of 14 open questions, relevant for locally convex spaces. Grothendieck introduced new methods, which allowed him to solve all these problems within a few months. In Nancy, he wrote his dissertation under those two professors on functional analysis, from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. From 1953 to 1955 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he refused to take French Nationality. By 1957, he set this subject aside in order to work in algebraic geometry and
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
. The same year he was invited to visit Harvard by Oscar Zariski, but the offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government, a position that, he was warned, might have landed him in prison. The prospect did not worry him, as long as he could have access to books. Comparing Grothendieck during his Nancy years to the École Normale Supérieure trained students at that time: Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, Bernard Malgrange, Leila Schneps says: His first works on topological vector spaces in 1953 have been successfully applied to physics and computer science, culminating in a relation between Grothendieck inequality and the EPR paradox, Einstein-Podolsky-Rosen paradox in quantum physics.


IHÉS years

In 1958, Grothendieck was installed at the Institut des hautes études scientifiques (IHÉS), a new privately funded research institute that, in effect, had been created for Jean Dieudonné and Grothendieck. Grothendieck attracted attention by an intense and highly productive activity of seminars there (''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. 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 Michael Artin (étale cohomology) and Nick Katz (monodromy theory and Lefschetz pencils). Jean Giraud (mathematician), Jean Giraud worked out torsor theory extensions of nonabelian cohomology. Many others like David Mumford, Robin Hartshorne, Barry Mazur and C.P. Ramanujam were also involved.


"Golden Age"

Alexander Grothendieck's work during the "Golden Age" period at the IHÉS established several unifying themes in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
, number theory, topology,
category theory Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...
and complex analysis. His first (pre-IHÉS) discovery in algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem, a generalisation of the Hirzebruch–Riemann–Roch theorem proved algebraically; in this context he also introduced
K-theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of scheme (mathematics), schemes, developing it in detail in his ''Éléments de géométrie algébrique'' (''EGA'') and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time. He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it. 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 category, derived categories, which were further developed by Verdier. The results of work on these and other topics were published in the ''EGA'' and in less polished form in the notes of the ''Séminaire de géométrie algébrique'' (''SGA'') that he directed at the IHÉS.


Political activism

Grothendieck's political views were Political radicalism, radical and pacifist, and he strongly opposed both United States Vietnam War, intervention in Vietnam and Soviet Empire, Soviet military expansionism. He gave lectures on
category theory Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...
in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam War. He retired from scientific life around 1970, having found out that IHÉS was partly funded by the military. He returned to academia a few years later as a professor at the
University of Montpellier The University of Montpellier (french: Université de Montpellier) is a French public research university A research university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of highe ...
. While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from the IHÉS, those who knew him say that the causes of the rupture ran deeper. Pierre Cartier (mathematician), Pierre Cartier, a ''visiteur de longue durée'' ("long-term guest") at the IHÉS, wrote a piece about Grothendieck for a special volume published on the occasion of th
IHÉS's fortieth anniversary
The ''Grothendieck Festschrift'', published in 1990, was a three-volume collection of research papers to mark his sixtieth birthday in 1988. 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. As Cartier puts it, Grothendieck came to find Bures-sur-Yvette "''une cage dorée''" ("a gilded cage"). While Grothendieck was at the IHÉS, opposition to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a mandarin of the scientific world. In addition, after several years at the IHÉS, Grothendieck seemed to cast about for new intellectual interests. By the late 1960s, he had started to become interested in scientific areas outside mathematics. David Ruelle, a physicist who joined the IHÉS faculty in 1964, said that Grothendieck came to talk to him a few times about physics.Ruelle invented the concept of a strange attractor in a dynamical system and, with the Dutch mathematician Floris Takens, produced a new model for turbulence during the 1970s. Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics. In 1970, 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. Grothendieck devoted the next three years to this group and served as the main editor of its bulletin. Although Grothendieck continued with mathematical enquiries his standard mathematical career, for the most part, ended when he left the IHÉS. After leaving the IHÉS Grothendieck became a temporary professor at Collège de France for two years. He then became a professor at the University of Montpellier, where he became increasingly estranged from the mathematical community. He formally retired in 1988, a few years after having accepted a research position at the Centre National de la Recherche Scientifique, CNRS.


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. Produced during 1980 and 1981, ''La Longue Marche à travers la théorie de Galois'' (''The Long March Through Galois Theory'') is a 1600-page handwritten manuscript containing many of the ideas that led to the ''Esquisse d'un programme''.Alexandre Grothendieck
Esquisse d'un ProgrammeEnglish translation
/ref> It also includes a study of Teichmüller space, Teichmüller theory. In 1983, stimulated by correspondence with Ronald Brown (mathematician), Ronald Brown and Tim Porter at Bangor University, Grothendieck wrote a 600-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, 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 Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
and prospects for a noncommutative theory of Stack (mathematics), 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 Fabien Morel and Vladimir Voevodsky in the mid-1990s. In 1984, Grothendieck wrote the proposal ''Esquisse d'un Programme'' ("Sketch of a Programme") for a position at the Centre National de la Recherche Scientifique (CNRS). 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. It was later published in the two-volum
''Geometric Galois Actions''
(Cambridge University Press, 1997). During this period, Grothendieck also gave his consent to publishing some of his drafts for EGA on Theorem of Bertini, Bertini-type theorems (''EGA'' V, published in Ulam Quarterly in 1992-1993 and later made available on th
Grothendieck Circle
web site in 2004). In the 1,000-page autobiographical manuscript ''Récoltes et semailles'' (1986) Grothendieck describes his approach to mathematics and his experiences in the mathematical community, a community that initially accepted him in an open and welcoming manner but which he progressively perceived to be governed by competition and status. 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. ''Récoltes et semailles'' work is now available on the internet in the French original, Alexander Grothendieck
et sémailles'', ''Réflexions et témoignage sur un passé de mathématicien.''"
and an English translation is underway. Parts of ''Récoltes et semailles'' have been translated into Spanish and into Russian and published in Moscow. In 1988 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 315-page manuscript written in 1987, is Grothendieck's account of how his consideration of the source of dreams led him to conclude that Existence of God, God exists. As part of the notes to this manuscript, Grothendieck described the life and work of 18 "mutants", people whom he admired as visionaries far ahead of their time and heralding a new age. The only mathematician on his list was Bernhard Riemann. Influenced by the Catholic mystic Marthe Robin who was claimed to survive on the Holy Eucharist alone, Grothendieck almost starved himself to death in 1988. His growing preoccupation with spiritual matters was also evident in a letter titled ''Lettre de la Bonne Nouvelle'' sent to 250 friends in January 1990. In it, he described his encounters with a deity and announced that a "New Age" would commence on 14 October 1996. Over 20,000 pages of Grothendieck's mathematical and other writings, held at the University of Montpellier, remain unpublished. They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal.


Retirement into reclusion and death

In 1991, Grothendieck moved to a new address which he did not provide to his previous contacts in the mathematical community. Very few people visited him afterward. Local villagers helped sustain him with a more varied diet after he tried to live on a staple of Taraxacum, dandelion soup. At some point, Leila Schneps and Pierre Lochak located him, then carried on a brief correspondence. Thus they became among "the last members of the mathematical establishment to come into contact with him". After his death, it was revealed that he lived alone in a house in Lasserre, Ariège, a small village at the foot of the
Pyrenees The Pyrenees (; es, Pirineos ; french: Pyrénées ; ca, Pirineus ; eu, Pirinioak ; oc, Pirenèus ; an, Pirineus) is a mountain range straddling the border of and . It extends nearly from its union with the to on the coast. It reaches a ma ...

Pyrenees
. In January 2010, Grothendieck wrote the letter "Déclaration d'intention de non-publication" to Luc Illusie, claiming that all materials published in his absence have been published without his permission. He asks that none of his work be reproduced in whole or in part and that copies of this work be removed from libraries. A website devoted to his work was called "an abomination." This order may have been reversed later in 2010. On 13 November 2014, aged 86, Grothendieck died in the hospital of Saint-Girons, Ariège.


Citizenship

Grothendieck was born in Weimar Republic, Weimar Germany. In 1938, aged ten, he moved to France as a refugee. Records of his nationality were destroyed in the fall of Germany in 1945 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. Part of this reluctance to hold French nationality is attributed to not wishing to serve in the French military, particularly due to the Algerian War (1954–62). He eventually applied for French citizenship in the early 1980s, well past the age that exempted him from military service.


Family

Grothendieck was very close to his mother to whom he dedicated his dissertation. She died in 1957 from the tuberculosis that she contracted in camps for displaced persons. He had five children: a son with his landlady during his time in Nancy, three children, Johanna (1959), Alexander (1961) and Mathieu (1965) with his wife Mireille Dufour, and one child with Justine Skalba, with whom he lived in a commune in the early 1970s.


Mathematical work

Grothendieck's early mathematical work was in functional analysis. Between 1949 and 1953 he worked on his doctoral thesis in this subject at Nancy, France, Nancy, supervised by Jean Dieudonné and Laurent Schwartz. 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 Lp space, Lp spaces in studying linear maps between topological vector spaces. In a few years, he had turned himself into a leading authority on this area of functional analysis—to the extent that Dieudonné compares his impact in this field to that of Stefan Banach, Banach. It is, however, in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
and related fields where Grothendieck did his most important and influential work. From about 1955 he started to work on sheaf (mathematics), sheaf theory and
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
, producing the influential "Tôhoku paper" (''Sur quelques points d'algèbre homologique'', published in the Tohoku Mathematical Journal in 1957) where he introduced Abelian category, abelian categories and applied their theory to show that sheaf cohomology can be defined as certain derived functors in this context. 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 ''Grothendieck's relative point of view, relative point of view'' (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems. The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite-dimensional; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a one-point space. In 1956, he applied the same thinking to the Riemann–Roch theorem, which had already recently been generalized to any dimension by Friedrich Hirzebruch, 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. This result was his first work in algebraic geometry. 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 1958 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 scheme (mathematics), 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 Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
, 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. 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. His 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.)


''EGA'', ''SGA'', ''FGA''

The bulk of Grothendieck's published work is collected in the monumental, 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. Grothendieck's work includes the invention of the Étale cohomology, étale and l-adic cohomology theories, which explain an observation of André Weil's that there is a 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, the last of which was settled by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.


Major mathematical contributions

In Grothendieck's retrospective ''Récoltes et Semailles'', he identified twelve of his contributions which he believed qualified as "great ideas". In chronological order, they are: # Topological tensor products and nuclear spaces. # "Continuous" and "discrete" Duality (mathematics), duality (derived category, derived categories, "six operations (mathematics), six operations"). # Yoga of the Grothendieck–Riemann–Roch theorem (
K-theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, relation with intersection theory). # Scheme (mathematics), Schemes. # Topos, Topoi. # Étale cohomology and l-adic cohomology. # Motive (algebraic geometry), Motives and the motivic Galois group (Grothendieck ⊗-categories). # Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients", ... # "Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra. # Tame topology. # Yoga of anabelian geometry, anabelian algebraic geometry, Galois–Teichmüller theory. # "Schematic" or "arithmetic" point of view for regular polyhedron, regular polyhedra and regular configurations of all kinds. Here the term ''yoga'' denotes a kind of "meta-theory" that can be used heuristically; Michel Raynaud writes the other terms "Ariadne's thread" and "philosophy" as effective equivalents. Grothendieck wrote that, of these themes, the largest in scope was topoi, as they synthesized algebraic geometry, topology, and arithmetic. The theme that had been most extensively developed was schemes, which were the framework "''par excellence''" for eight of the other themes (all but 1, 5, and 12). Grothendieck wrote that the first and last themes, topological tensor products and regular configurations, were of more modest size than the others. Topological tensor products had played the role of a tool rather than a source of inspiration for further developments; but he expected that regular configurations could not be exhausted within the lifetime of a mathematician who devoted himself to it. He believed that the deepest themes were motives, anabelian geometry, and Galois–Teichmüller theory.


Influence

Grothendieck is considered by many to be the greatest mathematician of the 20th century. In an obituary David Mumford and John Tate wrote:
Although mathematics became more and more abstract and general throughout the 20th century, it was Alexander Grothendieck who was the greatest master of this trend. His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves–and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed.Alexander Grothendieck obituary by David Mumford and John Tate
David Mumford at Brown and Harvard Universities: Archive for Reprints: ''Can one explain schemes to biologists'', 14 December 2014
By the 1970s, Grothendieck's work was seen as influential not only in algebraic geometry, and the allied fields of sheaf theory and homological algebra, but influenced logic, in the field of categorical logic.


Geometry

Grothendieck approached algebraic geometry by clarifying the foundations of the field, and by developing mathematical tools intended to prove a number of notable conjectures. Algebraic geometry has traditionally meant the understanding of geometric objects, such as algebraic curves and surfaces, through the study of the algebraic equations for those objects. Properties of algebraic equations are in turn studied using the techniques of ring theory. In this approach, the properties of a geometric object are related to the properties of an associated ring. The space (e.g., real, complex, or projective) in which the object is defined is extrinsic to the object, while the ring is intrinsic. Grothendieck laid a new foundation for algebraic geometry by making intrinsic spaces ("spectra") and associated rings the primary objects of study. To that end he developed the theory of Scheme (mathematics), schemes, which can be informally thought of as topological spaces on which a commutative ring is associated to every open subset of the space. Schemes have become the basic objects of study for practitioners of modern algebraic geometry. Their use as a foundation allowed geometry to absorb technical advances from other fields. His Grothendieck–Hirzebruch–Riemann–Roch theorem, generalization of the classical Riemann-Roch theorem related topological properties of complex algebraic curves to their algebraic structure. The tools he developed to prove this theorem started the study of Algebraic K-theory, algebraic and topological K-theory, which study the topological properties of objects by associating them with rings. Topological K-theory was founded by Michael Atiyah and Friedrich Hirzebruch, after direct contact with Grothendieck's ideas at the Bonn Arbeitstagung.


Cohomology theories

Grothendieck's construction of new cohomology theories, which use algebraic techniques to study topological objects, has influenced the development of algebraic number theory, algebraic topology, and representation theory. As part of this project, his creation of topos theory, a category-theoretic generalization of point-set topology, has influenced the fields of set theory and mathematical logic. The Weil conjectures were formulated in the later 1940s as a set of mathematical problems in arithmetic geometry. They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension. Grothendieck's discovery of the étale cohomology, ℓ-adic étale cohomology, the first example of a Weil cohomology theory, opened the way for a proof of the Weil conjectures, ultimately completed in the 1970s by his student Pierre Deligne. Grothendieck's large-scale approach has been called a "visionary program." The ℓ-adic cohomology then became a fundamental tool for number theorists, with applications to the Langlands program. Grothendieck's conjectural theory of Motive (algebraic geometry), motives was intended to be the "ℓ-adic" theory but without the choice of "ℓ", a prime number. It did not provide the intended route to the Weil conjectures, but has been behind modern developments in algebraic K-theory, Motivic cohomology, motivic homotopy theory, and motivic integration. This theory, Daniel Quillen's work, and Grothendieck's theory of Chern classes, are considered the background to the theory of algebraic cobordism, another algebraic analogue of topological ideas.


Category theory

Grothendieck's emphasis on the role of Universal property, universal properties across varied mathematical structures brought
category theory Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...
into the mainstream as an organizing principle for mathematics in general. Among its uses, category theory creates a common language for describing similar structures and techniques seen in many different mathematical systems. His notion of abelian category is now the basic object of study in
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
. The emergence of a separate mathematical discipline of category theory has been attributed to Grothendieck's influence, though unintentional.


In popular culture

The novel ''Colonel Lágrimas'' (''Colonel Tears'' in English, available by Restless Books) by Puerto Rican - Costa Rican writer Carlos Fonseca is a semibiographic novel about Grothendieck.


Publications

* *


See also

* * * *


Notes


References


Sources and further reading

* * * ** * * * * * * * * * * * Three-volume biography, first volume available in English, . ** * *


External links

* *
Séminaire Grothendieck
is a peripatetic seminar on Grothendieck view not just on mathematics
Grothendieck Circle
collection of mathematical and biographical information, photos, links to his writings

This is an account of how 'Pursuing Stacks' was written in response to a correspondence in English with Ronnie Brown an
Tim Porter
at Bangor, which continued until 1991. See als
Alexander Grothendieck: some recollections

Récoltes et Semailles

"Récoltes et Semailles" et "La Clef des Songes"
French originals and Spanish translations
English summary of "La Clef des Songes"

Video of a lecture
with photos from Grothendieck's life, given by Winfried Scharlau at IHES in 2009

—biographical sketch of Grothendieck by David Mumford & John Tate
Archives Grothendieck
*
Who Is Alexander Grothendieck?
Winfried Scharlau, Notices of the AMS 55(8), 2008. *
Alexander Grothendieck: A Country Known Only by Name
Pierre Cartier, Notices of the AMS 62(4), 2015.
Alexandre Grothendieck 1928–2014, Part 1
Notices of the AMS 63(3), 2016.
A. Grothendieck
by Mateo Carmona

* Semen Samsonovich Kutateladze, Kutateladze S.S.]
Rebelious Genius: In Memory of Alexander Grothendieck


* [https://www.lemonde.fr/sciences/article/2019/05/06/les-archives-insaisissables-d-alexandre-grothendieck_5459049_1650684.html Les-archives-insaisissables-d-alexandre-grothendieck] {{DEFAULTSORT:Grothendieck, Alexander Nicolas Bourbaki Algebraic geometers Algebraists Functional analysts Operator theorists Scientists from Berlin 20th-century French mathematicians Emigrants from Nazi Germany to France Stateless people Fields Medalists 1928 births 2014 deaths German people of Ukrainian-Jewish descent Nancy-Université alumni French pacifists