Alexander Grothendieck

Alexander Grothendieck
Grothendieck in Montréal (1970)
Alexander Grothendieck
Born	Alexander Grothendieck 28 3, 1928
Birthplace	Berlin, Prussia, Germany
Died	Template:Death date and age Saint-Lizier, Ariège, France
Nationality	German (by birth), French (from 1971)
Occupation	Mathematician
Known for	Renewing algebraic geometry; synthesis of algebraic geometry with number theory and topology
Education	PhD, University of Nancy (1953)
Awards	Fields Medal (1966), Émile Picard Medal, Crafoord Prize (1988, declined)

Alexander Grothendieck (28 March 1928 – 13 November 2014) was a German-born French mathematician whose profound contributions to algebraic geometry, homological algebra, and category theory reshaped the landscape of modern mathematics. Born in Berlin to politically radical parents, Grothendieck survived a childhood scarred by the upheavals of Nazism and the Second World War, spending years in internment camps in France before emerging as one of the most original mathematical thinkers of his era. He is considered by many mathematicians and historians of the discipline to be the greatest mathematician of the twentieth century. His work at the Institut des hautes études scientifiques (IHÉS) between 1958 and 1970 produced a torrent of ideas that fundamentally extended the scope of algebraic geometry, incorporating elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations. He received the Fields Medal in 1966 and was awarded the Crafoord Prize in 1988, which he declined. In the later decades of his life, Grothendieck withdrew from the mathematical community, eventually living as a recluse in a small village in the French Pyrenees until his death at the age of eighty-six.

Early Life

Alexander Grothendieck was born on 28 March 1928 in Berlin, Germany. His father, Alexander "Sascha" Schapiro (also known as Tanaroff), was a Russian Jew and anarchist revolutionary who had participated in various uprisings and spent years in Tsarist prisons. His mother, Hanka Grothendieck, was a German journalist and writer with radical political views. Both parents were deeply involved in left-wing politics, and Alexander took his mother's surname.

The political turmoil of the 1930s profoundly affected the family. When the Nazis came to power in Germany in 1933, Grothendieck's father fled to France, and his mother followed shortly thereafter, leaving the young Alexander in the care of a foster family in Hamburg, Germany. He lived with Wilhelm Heydorn, a Lutheran pastor and his wife, who raised him for several years while his parents were involved in the Spanish Civil War and other political struggles.

In 1939, Grothendieck was reunited with his mother in France, but the reunion came at a perilous time. With the German occupation of France, the family's situation became dangerous. His father, as a stateless Jew, was interned in the French camp at Le Vernet and was subsequently handed over to the Nazis, who deported him to Auschwitz, where he was killed in 1942. Meanwhile, Alexander and his mother were interned in the camp at Rieucros, near Mende, in southern France. Later, Alexander was placed in various homes and attended school at the Collège Cévenol in Chambon-sur-Lignon, a village in the Massif Central region that was known for sheltering refugees during the war.

These traumatic early experiences — the loss of his father, the years of displacement, the constant threat of persecution — left an enduring mark on Grothendieck. Despite the hardship, he managed to continue his education and showed early signs of exceptional mathematical talent. After the liberation of France, Grothendieck enrolled at the University of Montpellier, where he studied mathematics as an undergraduate.

Education

At the University of Montpellier, Grothendieck pursued undergraduate studies in mathematics between 1945 and 1948. According to later accounts, he found the teaching there unsatisfying and independently redeveloped substantial portions of measure theory and the Lebesgue integral, not knowing that the work had already been done. This early episode demonstrated both his remarkable originality and his tendency to work from first principles rather than building on existing literature.

In 1948, Grothendieck moved to Paris, where he attended seminars at the École Normale Supérieure and came into contact with leading French mathematicians including Henri Cartan and André Weil. On the recommendation of Cartan and Weil, he was directed to the University of Nancy to work with Laurent Schwartz and Jean Dieudonné, both prominent analysts. Under their supervision, Grothendieck completed his doctoral thesis, Produits tensoriels topologiques et espaces nucléaires (Topological Tensor Products and Nuclear Spaces), in 1953. This work, in the field of functional analysis, was itself a major achievement. Grothendieck introduced the concept of nuclear spaces and resolved several outstanding problems in the theory of topological vector spaces. The thesis established his reputation as a mathematician of extraordinary power even before he turned to the algebraic geometry that would define his career.

Career

Early Work in Functional Analysis (1949–1955)

Grothendieck began his productive career as a mathematician in 1949, initially working in functional analysis. His doctoral work at Nancy on topological tensor products was groundbreaking in its own right and would have been sufficient to establish a distinguished career. He produced a series of papers that made fundamental contributions to the theory of topological vector spaces, introducing concepts and techniques that remain central to the field. During this period, he also spent time at the University of São Paulo in Brazil and at the University of Kansas in the United States, pursuing visiting appointments as a stateless person who did not yet hold French citizenship.

Despite his considerable achievements in functional analysis, Grothendieck made a dramatic shift in the mid-1950s. He turned his attention to algebraic geometry and homological algebra, fields that he would transform more radically than any single mathematician before or since.

IHÉS and the Revolution in Algebraic Geometry (1958–1970)

In 1958, Grothendieck was appointed as a research professor at the newly founded Institut des hautes études scientifiques (IHÉS) in Bures-sur-Yvette, near Paris. This appointment marked the beginning of what is often described as the most productive period in twentieth-century mathematics. Over the next twelve years, Grothendieck led a monumental effort to rebuild the foundations of algebraic geometry from the ground up.

The centerpiece of this effort was the Éléments de géométrie algébrique (EGA), written in collaboration with Jean Dieudonné, and the Séminaire de Géométrie Algébrique du Bois Marie (SGA), a series of seminar notes produced with numerous collaborators and students. These works introduced the language of schemes, which generalized the classical notion of algebraic varieties and provided a unified framework capable of treating geometry over arbitrary commutative rings, including the integers. This "relative" point of view — in which geometric objects are studied not in isolation but in families parameterized by other geometric objects — proved extraordinarily fertile and led to advances across pure mathematics.

Grothendieck's approach was characterized by an emphasis on generality and abstraction. Rather than solving individual problems by ingenious tricks, he sought to develop theories of such depth and scope that difficult problems would become easy, or even trivial, consequences of the general framework. This philosophy, which he sometimes described as allowing the "rising sea" of theory to submerge the difficulties, was strikingly effective.

Among the specific contributions Grothendieck made during this period were the development of étale cohomology, which provided the tools necessary for the eventual proof of the Weil conjectures (completed by his student Pierre Deligne in 1974); the introduction of K-theory in algebraic geometry; the formulation and proof of the Grothendieck–Riemann–Roch theorem; the theory of schemes and their morphisms; topos theory; the theory of motives; and fundamental advances in the theory of descent and formal geometry. The sheer volume and depth of this output was staggering — the collected works from the IHÉS period filled thousands of pages and engaged dozens of collaborators and students.

Grothendieck attracted and mentored a remarkable group of doctoral students during this period, including Pierre Deligne, Luc Illusie, Jean-Louis Verdier, Michel Raynaud, Michèle Raynaud, Jean-Pierre Jouanolou, William Messing, Neantro Saavedra-Rivano, and Hoàng Xuân Sính, many of whom went on to distinguished careers of their own.

In 1966, Grothendieck was awarded the Fields Medal at the International Congress of Mathematicians in Moscow for his advances in algebraic geometry, homological algebra, and K-theory. However, he refused to travel to Moscow to accept the prize in person, partly as a protest against the Soviet military actions and political repression at the time.

Departure from IHÉS and Political Activism (1970–1972)

In 1970, Grothendieck left the IHÉS after discovering that a small portion of the institute's funding came from military sources (the French Ministry of Defense, known as DGRST). This discovery precipitated a crisis for Grothendieck, whose upbringing and personal history had left him with deep-seated opposition to militarism and political violence. He made his continued presence at IHÉS conditional on the removal of military funding, and when this condition was not met, he resigned.

His departure from IHÉS marked a turning point. Grothendieck became increasingly involved in political activism, particularly related to pacifism and environmentalism. He co-founded the environmentalist group Survivre et Vivre (Survive and Live), which was concerned with nuclear proliferation, environmental degradation, and questions of scientific responsibility. He gave lectures and talks on these subjects and sought to engage the scientific community in addressing what he considered existential threats to humanity.

University of Montpellier and Later Mathematical Work (1972–1988)

After leaving IHÉS, Grothendieck accepted a position at the University of Montpellier, where he taught from 1973 onward. Although he continued to produce mathematical work during this period, his output was more sporadic and unconventional than during his years at IHÉS. He increasingly worked in isolation and became more reluctant to publish through conventional channels.

Among the notable mathematical works from this later period was Esquisse d'un Programme (Sketch of a Programme), written in 1984 as part of a research proposal. In this document, Grothendieck outlined a sweeping vision for future mathematical research, introducing ideas including dessins d'enfants (children's drawings), a combinatorial approach to understanding Galois groups through their action on certain graphs drawn on surfaces, and the concept of an "anabelian geometry" in which topological and arithmetic information could be recovered from fundamental groups. The Esquisse also explored the notion of Teichmüller towers and the connections between number theory, topology, and algebraic geometry at the deepest structural level. Though it was not formally published at the time, the document circulated among mathematicians and has continued to inspire research for decades.

Another significant work from this period was Pursuing Stacks (1983), a long manuscript in the form of a letter to Daniel Quillen exploring higher-dimensional category theory and homotopical algebra. Like the Esquisse, this work anticipated developments that would not come to fruition until years later, and it remains influential in current research on higher categories and derived algebraic geometry.

In 1988, the Royal Swedish Academy of Sciences awarded Grothendieck the Crafoord Prize, jointly with Pierre Deligne. Grothendieck declined the prize, writing a public letter in which he explained his reasons. He stated that he did not need the prize money and that the award system did not serve the interests of mathematics or scientists. He also expressed concern about what he perceived as ethical failings in the mathematical community.

Récoltes et Semailles and Withdrawal (1985–1991)

Between 1985 and 1987, Grothendieck composed Récoltes et Semailles (Reaping and Sowing), a massive autobiographical and philosophical text running to over a thousand pages. In this work, Grothendieck reflected on his mathematical career, his relationships with colleagues and students, and what he saw as the misappropriation and misattribution of his ideas by the mathematical establishment. The text is both a meditation on the creative process in mathematics and a sometimes bitter critique of the professional norms of academic life. It was circulated privately and has been extensively discussed by historians of mathematics and by mathematicians who knew Grothendieck personally.

Grothendieck retired from the University of Montpellier in 1988. In 1991, he left his home abruptly and moved to the village of Lasserre in the Ariège department of the French Pyrenees, where he lived in near-total seclusion for the remaining twenty-three years of his life. He cut off almost all contact with the mathematical world and with former colleagues and friends. During this period, he devoted himself to writing — producing thousands of pages of manuscripts on mathematics, philosophy, and spirituality — but he did not allow any of this work to be published.

Personal Life

Grothendieck's personal life was marked by the same intensity and unconventionality that characterized his mathematics. He was stateless for much of his early adult life, holding a Nansen passport issued to stateless persons, and did not become a French citizen until 1971.

He had several long-term relationships and fathered at least five children, but never married in a conventional sense. His personal relationships were often complicated by his single-minded devotion to his work and, later, by his increasing withdrawal from society.

Grothendieck's political and ethical convictions were deeply held and influenced his professional decisions throughout his career. His opposition to military funding led to his departure from IHÉS, and his environmental activism through Survivre et Vivre reflected a genuine commitment to pacifism and ecological responsibility that predated the mainstream environmental movement.

In his later years, Grothendieck turned to spiritual and religious pursuits. He explored Buddhism for a time before developing a more Catholic Christian orientation, though his spirituality was highly personal and idiosyncratic. During his years in Lasserre, he lived an austere and solitary life, rarely venturing out and receiving almost no visitors. He reportedly continued to write prolifically, filling boxes with manuscripts on mathematics, metaphysics, and theology.

Alexander Grothendieck died on 13 November 2014 in the hospital of Saint-Lizier, Ariège, France, at the age of eighty-six.

Recognition

Grothendieck received some of the highest honors available to a mathematician. In 1966, he was awarded the Fields Medal, often described as the most prestigious prize in mathematics, for his transformative contributions to algebraic geometry, homological algebra, and K-theory.

In 1977, he received the Émile Picard Medal from the Académie des Sciences in Paris.

In 1988, the Royal Swedish Academy of Sciences awarded him the Crafoord Prize jointly with Pierre Deligne. The Crafoord Prize was established specifically to recognize contributions in scientific fields not covered by the Nobel Prizes, including mathematics. Grothendieck declined the award, writing a widely circulated letter explaining his decision and criticizing the culture of prizes in academia.

Despite his withdrawal from public life, Grothendieck's reputation continued to grow in the decades after his active career. His former students and collaborators carried his ideas forward, and the frameworks he developed — particularly the theory of schemes, étale cohomology, and topos theory — became the standard language of modern algebraic geometry.

After his death in 2014, the University of Montpellier established an archive of Grothendieck's mathematical writings, and efforts were made to catalogue and eventually publish the vast body of unpublished manuscripts he left behind.

In 2024, reporting by The Guardian highlighted renewed interest in Grothendieck's later, unpublished writings, with some researchers speculating that ideas contained in his metaphysical manuscripts might have applications to artificial intelligence and other contemporary fields.

Legacy

Alexander Grothendieck's influence on mathematics is difficult to overstate. He fundamentally reshaped algebraic geometry, transforming it from a field concerned primarily with the study of curves and surfaces defined by polynomial equations into a vast, interconnected discipline that incorporated tools and insights from commutative algebra, homological algebra, topology, number theory, and category theory. The language and concepts he introduced — schemes, toposes, étale cohomology, motives, descent, stacks — are now the everyday working tools of algebraic geometers, number theorists, and many other mathematicians.

The "relative" point of view that Grothendieck championed — studying mathematical objects not in isolation but through their relationships to other objects — has become a pervasive organizing principle in modern mathematics. This perspective was essential to the proof of the Weil conjectures by Pierre Deligne in 1974, one of the great achievements of twentieth-century mathematics, which relied on the étale cohomology theory that Grothendieck had developed.

Grothendieck's influence extended beyond his published work. His approach to mathematics — seeking maximal generality, building vast theoretical frameworks, and allowing the "right" definitions and structures to reveal the solutions to problems — has profoundly influenced mathematical culture and methodology. Many mathematicians who never met him have adopted his style of thinking, and his ideas have spawned entire subfields of mathematics, including derived algebraic geometry, motivic homotopy theory, and higher category theory.

His life story — from the stateless child of radical parents, surviving internment and the loss of his father in the Holocaust, to the most creative mathematician of his generation, and finally to the hermit of Lasserre — has also captured the imagination of the broader public and has been the subject of numerous biographical accounts, documentaries, and public lectures.

The extensive mathematical archives he left behind — housed at the University of Montpellier and currently being catalogued and studied — may yet yield further insights, ensuring that Grothendieck's influence will continue to unfold for years to come.