AnarWiki/markdown/Alexander_Grothendieck.md

**Alexander Grothendieck** (/ˈɡroʊtəndiːk/; <small>German:
</small>\[ˈɡroːtn̩diːk\]; <small>French: </small>\[ɡʁɔtɛndik\]; 28
March 1928 – 13 November 2014) was a French mathematician who became the
leading figure in the creation of modern algebraic
geometry.<sup>\[6\]\[7\]</sup> His research extended the scope of the
field and added elements of commutative algebra, homological algebra,
sheaf theory and category theory to its foundations, while his so-called
"relative" perspective led to revolutionary advances in many areas of
pure mathematics.<sup>\[6\]\[8\]</sup> He is considered by many to be
the greatest mathematician of the 20th century.<sup>\[9\]</sup>

Born in Germany, Grothendieck was raised and lived primarily in France.
For much of his working life, however, he was, in effect,
stateless.<sup>\[1\]</sup> As he consistently spelled his first name
"Alexander" rather than "Alexandre"<sup>\[10\]</sup> and his surname,
taken from his mother, was the Dutch-like Low German "Grothendieck", he
was sometimes mistakenly believed to be of Dutch
origin.<sup>\[11\]</sup>

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 later became professor at the
University of Montpellier<sup>\[4\]</sup> and, while still producing
relevant mathematical work, he withdrew from the mathematical community
and devoted himself to political causes. Soon after his formal
retirement in 1988, he moved to the French village of Lasserre in
Pyrenees, where he lived secluded, still working tirelessly on
mathematics until his death in 2014.<sup>\[12\]</sup>

## Life

### Family and childhood

Grothendieck was born in Berlin to anarchist parents. His father,
[Alexander "Sascha" Schapiro](Sascha_Schapiro "wikilink") (also known as
Alexander Tanaroff), had Hasidic Jewish roots and had been imprisoned in
Russia before moving to Germany in 1922, while his mother, Johanna
"Hanka" Grothendieck, came from a Protestant family in Hamburg and
worked as a journalist. Both had broken away from their early
backgrounds in their teens.<sup>\[13\]</sup> At the time of his birth,
Grothendieck's mother was married to the journalist Johannes Raddatz and
his birthname was initially recorded as "Alexander Raddatz." The
marriage was dissolved in 1929 and Schapiro/Tanaroff acknowledged his
paternity, but never married Hanka.<sup>\[13\]</sup>

Grothendieck lived with his parents in Berlin until the end of 1933,
when his father moved to Paris to evade Nazism, followed soon thereafter
by his mother. They left Grothendieck in the care of Wilhelm Heydorn, a
Lutheran pastor and teacher<sup>\[14\]</sup> <sup>\[15\]</sup> in
Hamburg. During this time, his parents took part in the Spanish Civil
War, according to Winfried Scharlau<sup> \[de\]</sup>, as non-combatant
auxiliaries,<sup>\[16\]</sup> though others state that Sascha fought in
the anarchist militia.<sup>\[17\]</sup>

### World War II

In May 1939, Grothendieck was put on a train in Hamburg for France.
Shortly afterwards his father was interned in Le
Vernet.<sup>\[18\]</sup> He and his mother were then interned in various
camps from 1940 to 1942 as "undesirable dangerous
foreigners".<sup>\[19\]</sup> 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. Once
Alexander managed to escape from the camp, intending to assassinate
Hitler.<sup>\[18\]</sup> Later, his mother Hanka was transferred to the
Gurs internment camp for the remainder of World War II.<sup>\[18\]</sup>
Alexander was permitted to live, separated from his
mother,<sup>\[20\]</sup> in the village of Le Chambon-sur-Lignon,
sheltered and hidden in local boarding houses or pensions, though he
occasionally had to seek refuge in the woods during Nazis raids,
surviving at times without food or water for several
days.<sup>\[18\]\[20\]</sup> His father was arrested under the Vichy
anti-Jewish legislation, and sent to the Drancy, and then handed over by
the French Vichy government to the Germans to be sent to be murdered at
the Auschwitz concentration camp in 1942.<sup>\[7\]\[21\]</sup> 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.<sup>\[22\]</sup>

### Studies and contact with research mathematics

After the war, the young Grothendieck studied mathematics in France,
initially at the University of Montpellier where he did not initially
perform well, failing such classes as astronomy.<sup>\[23\]</sup>
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.<sup>\[24\]</sup>

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 he wrote his dissertation under Laurent
Schwartz and Jean Dieudonné on functional analysis, from 1950 to
1953.<sup>\[25\]</sup> At this time he was a leading expert in the
theory of topological vector spaces.<sup>\[26\]</sup> 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.<sup>\[25\]</sup> 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.<sup>\[27\]</sup>

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:

### 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.<sup>\[29\]</sup> 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).<sup>\[14\]</sup>
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.<sup>\[30\]</sup>

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 worked out torsor theory
extensions of nonabelian cohomology. Many others were involved.

### "Golden Age"

Alexander Grothendieck's work during the "Golden Age" period at the IHÉS
established several unifying themes in algebraic geometry, number
theory, topology, category theory and complex analysis.<sup>\[25\]</sup>
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. Then, following the programme he outlined in
his talk at the 1958 International Congress of Mathematicians, he
introduced the theory of 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.<sup>\[14\]</sup> 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
categories, which were further developed by Verdier.<sup>\[31\]</sup>

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.<sup>\[14\]</sup>

### Political activism

Grothendieck's political views were radical and pacifist, and he
strongly opposed both United States intervention in Vietnam and Soviet
military expansionism. He gave lectures on category theory in the
forests surrounding Hanoi while the city was being bombed, to protest
against the Vietnam War.<sup>\[32\]</sup> He retired from scientific
life around 1970, having found out that IHÉS was partly funded by the
military.<sup>\[33\]</sup> He returned to academia a few years later as
a professor at the University of Montpellier.

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, 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 the 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.<sup>\[34\]</sup>

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.<sup>\[29\]</sup> 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.<sup>\[n 1\]</sup> Biology interested Grothendieck much
more than physics, and he organized some seminars on biological
topics.<sup>\[35\]</sup>

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.<sup>\[36\]</sup> Grothendieck devoted the next three years
to this group and served as the main editor of its
bulletin.<sup>\[4\]</sup>

After leaving the IHÉS, Grothendieck became a temporary professor at
Collège de France for two years.<sup>\[36\]</sup> He then became a
professor at the University of Montpellier, where he became increasingly
estranged from the mathematical community. His mathematical career, for
the most part, ended when he left the IHÉS.<sup>\[7\]</sup> He formally
retired in 1988, a few years after having accepted a research position
at the CNRS.<sup>\[4\]</sup>

### 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*.<sup>\[37\]</sup> It also includes a study of
Teichmüller theory.

In 1983, stimulated by correspondence with 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 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 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*.<sup>\[6\]</sup> Much of this work anticipated the
subsequent development of the motivic homotopy theory of Fabien Morel
and V. Voevodsky in the mid-1990s.

In 1984, Grothendieck wrote the proposal *Esquisse d'un
Programme*<sup>\[37\]</sup> ("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-volume *Geometric Galois Actions* (Cambridge University Press,
1997).

During this period, Grothendieck also gave his consent to publishing
some of his drafts for EGA on Bertini-type theorems (*EGA* V, published
in Ulam Quarterly in 1992-1993 and later made available on the
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.<sup>\[14\]</sup> *Récoltes et semailles* work is now
available on the internet in the French original,<sup>\[38\]</sup> and
an English translation is underway. Parts of *Récoltes et semailles*
have been translated into Spanish<sup>\[39\]</sup> and into Russian and
published in Moscow.<sup>\[40\]</sup>

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."<sup>\[41\]</sup>

*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 God exists.<sup>\[42\]</sup> 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.<sup>\[43\]</sup> The only
mathematician on his list was Bernhard Riemann.<sup>\[44\]</sup>
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.<sup>\[4\]</sup> 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.<sup>\[6\]</sup>

Over 20,000 pages of Grothendieck's mathematical and other writings,
held at the University of Montpellier, remain
unpublished.<sup>\[45\]</sup> They have been digitized for preservation
and are freely available in open access through the Institut
Montpelliérain Alexander Grothendieck portal.<sup>\[46\]</sup>

### 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.<sup>\[4\]</sup>
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
dandelion soup.<sup>\[47\]</sup> 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.<sup>\[48\]</sup>

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.<sup>\[49\]</sup> A
website devoted to his work was called "an
abomination."<sup>\[50\]</sup> This order may have been reversed later
in 2010.<sup>\[51\]</sup>

On 13 November 2014, aged 86, Grothendieck died in the hospital of
Saint-Girons, Ariège.<sup>\[22\]\[52\]</sup>

### Citizenship

Grothendieck was born in 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.<sup>\[1\]\[2\]\[3\]</sup> 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).<sup>\[53\]\[29\]\[2\]</sup> He eventually applied for
French citizenship in the early 1980s, well past the age that exempted
him from military service.<sup>\[29\]</sup>

### 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.<sup>\[36\]</sup> He had five children: a
son with his landlady during his time in Nancy,<sup>\[29\]</sup> three
children, Johanna (1959), Alexander (1961) and Mathieu (1965) with his
wife Mireille Dufour,<sup>\[54\]\[4\]</sup> and one child with Justine
Skalba, with whom he lived in a commune in the early
1970s.<sup>\[4\]</sup>

## 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, 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 L<sup>p</sup> 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 Banach.<sup>\[55\]</sup>

It is, however, in algebraic geometry and related fields where
Grothendieck did his most important and influential work. From about
1955 he started to work on sheaf theory and homological algebra,
producing the influential "Tôhoku paper" (*Sur quelques points d'algèbre
homologique*, published in the Tohoku Mathematical Journal in 1957)
where he introduced abelian categories and applied their theory to show
that sheaf cohomology can be defined as certain derived functors in this
context.<sup>\[14\]</sup>

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 *relative point of view* (pairs of varieties related by a morphism),
allowing a broad generalization of many classical
theorems.<sup>\[36\]</sup> 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 Hirzebruch.
The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at
the initial Mathematische Arbeitstagung in Bonn, in
1957.<sup>\[36\]</sup> 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 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, and close analogues of the methods of algebraic topology, all
in an integrated way.<sup>\[14\]\[56\]\[57\]</sup>

He is also noted for his mastery of abstract approaches to mathematics
and his perfectionism in matters of formulation and
presentation.<sup>\[30\]</sup> 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.)<sup>\[58\]\[59\]</sup>

### *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.<sup>\[14\]</sup>

Grothendieck's work includes the invention of the 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.<sup>\[36\]</sup> 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.<sup>\[14\]</sup>

### Major mathematical contributions

In Grothendieck's retrospective *Récoltes et Semailles*, he identified
twelve of his contributions which he believed qualified as "great
ideas".<sup>\[60\]</sup> In chronological order, they are:

1.  Topological tensor products and nuclear spaces.
2.  "Continuous" and "discrete" duality (derived categories, "six
    operations").
3.  Yoga of the Grothendieck–Riemann–Roch theorem (K-theory, relation
    with intersection theory).
4.  Schemes.
5.  Topoi.
6.  Étale cohomology and l-adic cohomology.
7.  Motives and the motivic Galois group (Grothendieck ⊗-categories).
8.  Crystals and crystalline cohomology, yoga of "de Rham coefficients",
    "Hodge coefficients", ...
9.  "Topological algebra": ∞-stacks, derivators; cohomological formalism
    of topoi as inspiration for a new homotopical algebra.
10. Tame topology.
11. Yoga of anabelian algebraic geometry, Galois–Teichmüller theory.
12. "Schematic" or "arithmetic" point of view for 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.<sup>\[61\]</sup>

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.<sup>\[62\]</sup>

## Influence

Grothendieck is considered by many to be the greatest mathematician of
the 20th century.<sup>\[9\]</sup> In an obituary David Mumford and John
Tate wrote:

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,<sup>\[63\]</sup> but influenced logic, in the field
of categorical logic.<sup>\[64\]</sup>

### 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 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.<sup>\[65\]</sup>

His 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 and topological K-theory, which study the topological
properties of objects by associating them with rings.<sup>\[66\]</sup>
Topological K-theory was founded by Michael Atiyah, after direct contact
with Grothendieck's ideas at the Bonn Arbeitstagung.<sup>\[67\]</sup>

### 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.<sup>\[63\]</sup>

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 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.<sup>\[66\]</sup> Grothendieck's large-scale approach has
been called a "visionary program."<sup>\[68\]</sup> The ℓ-adic
cohomology then became a fundamental tool for number theorists, with
applications to the Langlands program.<sup>\[69\]</sup>

Grothendieck's conjectural theory of 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 homotopy
theory, and motivic integration.<sup>\[70\]</sup> 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.<sup>\[71\]</sup>

### Category theory

Grothendieck's emphasis on the role of universal properties across
varied mathematical structures brought category theory 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.<sup>\[72\]</sup> His notion of abelian category is now the
basic object of study in homological algebra.<sup>\[73\]</sup> The
emergence of a separate mathematical discipline of category theory has
been attributed to Grothendieck's influence, though
unintentional.<sup>\[74\]</sup>