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Alexander Grothendieck

Alexander Grothendieck (28 March 1928 – 13 November 2014) was a mathematician of exceptional influence, whose work reshaped the foundations of algebraic geometry and related fields through abstract categorical frameworks.^[1]^[2] Developing from self-taught origins amid wartime disruptions, he introduced key innovations including sheaf cohomology, the concept of schemes generalizing varieties to arbitrary commutative rings, étale cohomology for arithmetic applications, and topos theory extending geometry to logic and set theory.^[3]^[4] His 1957 Tohoku paper revolutionized homological algebra by emphasizing derived categories and functors, while his seminars at the Institut des Hautes Études Scientifiques (IHES) from 1958 produced Éléments de géométrie algébrique, a comprehensive reformulation enabling proofs of long-standing conjectures like the Riemann hypothesis for finite fields via the Weil conjectures.^[1] Awarded the Fields Medal in 1966 for these advances building on predecessors like Weil and Cartan, Grothendieck declined the honor's Moscow ceremony over Soviet policies but accepted the prize itself.^[3] In 1970, he resigned from IHES upon discovering its funding ties to French military research, urging colleagues to follow in protest; he then pursued environmental and pacifist activism through groups like Survivre before retreating to seclusion in the French Pyrenees around 1991, producing voluminous unpublished manuscripts on mathematics, philosophy, and mysticism until his death.^[5]^[6]

Early Life and Background

Family Origins and Childhood

Alexander Grothendieck was born on 28 March 1928 in Berlin, Germany, to parents engaged in radical political activities. His biological father, Alexander Schapiro (1890–1942), known as Sascha or under the alias Alexander Tanaroff, was a Russian Jew born in Novozybkov who fought in the 1905 Revolution against the Tsar, endured imprisonment, escaped during the 1917 Revolution, and lost his left arm amid the upheavals; he later worked as a street photographer in Berlin and Paris. His mother, Johanna "Hanka" Grothendieck (1900–1957), originated from Hamburg, Germany, belonged to a Protestant family, and participated in left-wing groups as a journalist; she had previously married Alfred Raddatz, with whom she had a daughter, Frode "Maidi" (born 1924), before divorcing in 1929. Schapiro and Hanka lived together without formal marriage, and Grothendieck was registered at birth as Alexander Raddatz but eventually took his mother's surname.^[6] The family lived in Berlin from 1928 to 1933, where Schapiro operated a photographic studio at Brunnenstrasse 165 in the Mitte district. With the Nazi ascent to power, Schapiro relocated to Paris in May 1933 to evade persecution, while Hanka and the young Grothendieck initially stayed; by January 1934, the five-year-old was placed in foster care with Wilhelm and Dagmar Heydorn in Hamburg, attending local schools and gymnasium until April 1939. Meanwhile, Hanka pursued anarchist involvements, traveling to Spain from 1936 to 1939 before returning to France in April 1939. Grothendieck reunited with his mother in Nîmes during the summer of 1939, immediately preceding the onset of World War II; this peripatetic existence, driven by his parents' activism and Europe's political instability, characterized his early childhood.^[6]

World War II and Displacement

The rise of the Nazi regime profoundly disrupted Grothendieck's early life. Adolf Hitler became Chancellor on January 30, 1933, followed by a boycott of Jewish businesses on April 1 and anti-Jewish laws enacted on April 7, targeting individuals like his father, Alexander "Sascha" Schapiro, a Russian-Jewish anarchist.^[6] Schapiro fled to Paris in May 1933, while Grothendieck's mother, Johanna "Hanka" Grothendieck, remained in Berlin with her son until December 1933.^[6] In January 1934, Hanka placed the five-year-old Grothendieck with foster parents, Wilhelm and Dagmar Heydorn, in Hamburg, where he resided until April 1939; meanwhile, she joined Schapiro in Paris and both participated in the Spanish Civil War from 1936 to 1939 before returning to France.^[6] Reunited with his mother in Nîmes during the summer of 1939, Grothendieck faced further upheaval with the outbreak of World War II in September and France's internment policies for foreigners deemed "enemy aliens."^[6] He and his mother were interned at the Rieucros Camp near Mende, established in January 1939 for female refugees and their children, under a November 1938 law targeting stateless persons.^[6]^[7] Conditions were harsh, marked by displacement, hunger, and poverty; Hanka contracted tuberculosis there, which contributed to her later death.^[6] Schapiro, stateless and Jewish, was separately interned at the Camp du Vernet d'Ariège.^[6] In 1942, as German forces occupied Vichy France, the Rieucros Camp closed, transferring inmates including Hanka to the Gurs internment camp near Pau in the Pyrenees; Grothendieck, then 13, was permitted to leave for schooling but lived in hiding to evade authorities.^[6]^[6] That August, the Vichy government handed Schapiro over to the Nazis, who deported him from Vernet to Auschwitz, where he perished.^[6] Grothendieck found refuge at Le Chambon-sur-Lignon, a Protestant village known for sheltering Jewish children, attending the Collège Cévenol and earning his baccalauréat in 1945 despite ongoing risks from Gestapo roundups.^[6] Hanka, released earlier, survived the war but in frail health, reuniting with her son postwar in Maisargues near Montpellier.^[6] These years of separation, internment, and loss instilled in Grothendieck a profound sense of statelessness and resilience amid the causal chain of Nazi persecution and wartime policies.^[6]

Education and Entry into Mathematics

Post-War Studies in France

Following the end of World War II, Grothendieck relocated with his mother to Maisargues near Montpellier and commenced mathematical studies at the University of Montpellier in 1945, supplementing a small scholarship by working in local vineyards to support himself.^[6] ^[8] During this period from 1945 to 1948, he pursued a licence ès mathématiques amid personal challenges, including his stateless status—which he maintained to avoid French military conscription—and initial dissatisfaction with the university's teaching quality, which he found inadequate for rigorous development.^[6] ^[8] Working in relative isolation, Grothendieck independently rediscovered a generalized version of Henri Lebesgue's integral theory while exploring measures, demonstrating early self-directed aptitude despite gaps in formal preparation from wartime disruptions.^[6] ^[8] In 1948, after obtaining his licence, Grothendieck traveled to Paris for a year, auditing courses at the École Normale Supérieure and engaging with advanced seminars, particularly Henri Cartan's on algebraic topology and sheaf theory, which introduced him to contemporary research frontiers.^[6] Key interactions there included figures like Cartan and André Weil, exposing him to influential ideas in topology and geometry that later shaped his work, though his focus remained on functional analysis.^[6] This brief Parisian interlude bridged his provincial studies to more sophisticated environments, highlighting his rapid assimilation of cutting-edge concepts despite lacking prior institutional prestige.^[6] By 1949, seeking stronger instruction in functional analysis, Grothendieck moved to the University of Nancy, where he resided with his mother—whose health had been compromised by tuberculosis—and pursued advanced studies leading to his doctorate.^[6] ^[9] Under the guidance of Jean Dieudonné locally and Laurent Schwartz as formal supervisor, he delved into topological vector spaces, contributing foundational results on tensor products and nuclear spaces amid Nancy's active mathematical seminars.^[6] ^[9] He completed his thèse de doctorat, titled Produits tensoriels topologiques et espaces nucléaires, in 1953, defending it on 28 February in Paris, which established him as an expert in the field and marked the culmination of his formative post-war training in France.^[6]

Early Research and Influences

Grothendieck commenced formal mathematical studies at the University of Montpellier in 1945, following his return to France after wartime displacement, but encountered limited advanced instruction there, prompting extensive self-study in areas such as topology and analysis.^[6] By 1948, dissatisfied with the local faculty, he relocated to Paris in December of that year and attended the influential seminar led by Henri Cartan at the Sorbonne, where exposure to contemporary French mathematics, including discussions on sheaves and cohomology, began shaping his perspective.^[10]^[11] Cartan, recognizing Grothendieck's potential, advised him to pursue doctoral work in Nancy with Laurent Schwartz, a leading figure in functional analysis known for his development of distribution theory.^[6] In Nancy from 1949 to 1953, Grothendieck worked under the nominal supervision of Schwartz and Jean Dieudonné, completing his doctoral thesis titled Produits tensoriels topologiques et espaces nucléaires in 1953, which systematically generalized Banach space theory to locally convex topological vector spaces.^[10]^[12] His early research, initiated with papers published from 1950 onward—such as "Une caractérisation vectorielle métrique des espaces L_p" (1950)—focused on metric characterizations of Lebesgue spaces, continuity of linear functionals, and the structure of tensor products, often independently rediscovering or extending results from predecessors like Stefan Banach and Frigyes Riesz while posing foundational questions about completeness and duality in infinite-dimensional settings.^[13] These contributions established rigorous frameworks for handling pathological behaviors in topological vector spaces, influencing subsequent developments in operator theory and partial differential equations.^[12] Key influences on Grothendieck's early work stemmed from the axiomatic rigor of the Nicolas Bourbaki collective, whose emphasis on structural abstraction aligned with his inclination toward unifying disparate concepts, as well as Schwartz's probabilistic and distributional approaches to analysis, which informed his handling of generalized functions.^[11] However, Grothendieck's rapid progress frequently outstripped his mentors; for instance, during his Nancy tenure, he developed the notion of nuclear spaces—spaces where integral representations of operators hold—anticipating applications in quantum field theory, with minimal direct guidance beyond seminar interactions.^[13] Cartan's topological seminars also exposed him to early ideas in algebraic topology, planting seeds for his later pivot toward homological methods, though his initial output remained anchored in analysis until the mid-1950s.^[6] This phase underscored Grothendieck's emergent style: deriving broad theorems from minimal axioms, often critiquing ad hoc classical techniques in favor of categorical generality.^[12]

Professional Career and Peak Productivity

Appointment at IHÉS

The Institut des Hautes Études Scientifiques (IHÉS) was established in 1958 by Léon Motchane, a Russian-born businessman with a passion for mathematics, who sought to create a French counterpart to the Institute for Advanced Study in Princeton.^[14] Motchane initially recruited Jean Dieudonné as the institute's first permanent professor in mathematics. Dieudonné agreed to join on the condition that Alexander Grothendieck be appointed alongside him, recognizing Grothendieck's emerging talent in algebraic geometry and related fields.^[15] In 1958, Grothendieck was thus appointed as a permanent professor at IHÉS, a position that provided him with stable funding and freedom from teaching obligations, allowing full dedication to research.^[15] ^[16] This appointment marked a pivotal shift, relocating him from temporary positions in Nancy and São Paulo to a dedicated research environment in Bures-sur-Yvette.^[6] The institute's early years were modest, operating initially without permanent facilities, yet this setup fostered an atmosphere of intense mathematical collaboration.^[16] Grothendieck's role at IHÉS quickly centered on leading seminars, such as the Séminaire de Géométrie Algébrique (SGA), which became instrumental in disseminating his ideas and training a generation of mathematicians.^[15] His appointment coincided with IHÉS's foundational phase, where he and Dieudonné formed the core of the mathematics section, emphasizing pure research over applied or institutional pressures.^[17] This period laid the groundwork for Grothendieck's transformative contributions, unencumbered by administrative duties.^[15]

The "Golden Age" of Contributions (1958–1970)

Upon his appointment as permanent professor at the Institut des Hautes Études Scientifiques (IHÉS) in 1958, Grothendieck entered a phase of extraordinary mathematical output that lasted until 1970, fundamentally redefining algebraic geometry through abstract, functorial methods.^[6] In collaboration with Jean Dieudonné, he produced the Éléments de géométrie algébrique (EGA), a foundational series with four volumes published in fascicles between 1960 and 1967, which introduced schemes as a unifying framework for studying algebraic varieties over arbitrary rings, resolving longstanding issues in classical algebraic geometry.^[6] This work emphasized relative notions and base change, enabling geometric interpretations across diverse base schemes.^[6] Grothendieck directed the Séminaire de Géométrie Algébrique (SGA) at IHÉS, a series of weekly seminars starting in 1960 that attracted leading mathematicians and generated multi-volume proceedings documenting cutting-edge developments.^[6] Key outcomes included SGA 1 (1960–1961) on étale coverings and fundamental groups, providing an algebraic analogue to topological fundamental groups for varieties, and SGA 4 (1963–1964) on étale cohomology, a cohomology theory suited to proving Weil's conjectures on zeta functions of varieties over finite fields.^[6] These seminars fostered a collaborative environment where Grothendieck's visionary ideas, often sketched rapidly and refined by participants, unified algebraic geometry with number theory, topology, and homological algebra.^[6] His innovations extended to category theory, with the development of topoi in the mid-1960s as generalized topological spaces via sites and sheaves, and to K-theory, where he defined Grothendieck groups for schemes, leading to the Grothendieck-Riemann-Roch theorem (published 1958 but expanded in this period).^[6] Working with intense focus—often 12 hours daily—Grothendieck produced thousands of pages of manuscripts, training a generation of geometers and establishing IHÉS as a global hub.^[10] In recognition, he received the Fields Medal in 1966 for advances in homological algebra, algebraic topology, and K-theory, though he declined to attend the Moscow ceremony in protest of Soviet policies.^[3]

Core Mathematical Innovations

Revolutionizing Algebraic Geometry

Grothendieck's approach to algebraic geometry emphasized abstraction and generality, shifting the field from its classical focus on varieties over algebraically closed fields to a framework accommodating arbitrary rings and relative situations.^[18] This revolution began in earnest around 1958, following his earlier work on sheaf cohomology in the 1957 Tohoku paper, as he sought to resolve foundational inconsistencies in handling infinitesimal structures and arithmetic geometry.^[19] By prioritizing functorial viewpoints and representable functors, he replaced ad hoc geometric intuitions with rigorous categorical methods, enabling precise treatment of moduli problems and deformation theory.^[20] Central to this transformation was the introduction of schemes in 1960, defined as locally ringed spaces that are locally affine, with affine schemes constructed as the spectrum Spec(R) of a commutative ring R, incorporating prime ideals as points and structure sheaves encoding ring-theoretic data.^[21] Schemes extended classical varieties by admitting nilpotent elements in structure sheaves, thus capturing "infinitesimal thickenings" absent in reduced schemes, and allowed base change over any scheme, facilitating relative geometry over non-fields like the integers.^[22] This framework, systematically developed in the multi-volume Éléments de géométrie algébrique (EGA), co-edited with Jean Dieudonné and published starting in 1960, provided a comprehensive foundation integrating commutative algebra, topology, and geometry.^[23] The EGA and associated Séminaire de Géométrie Algébrique (SGA) seminars, conducted at IHÉS from 1960 to 1969, disseminated these ideas through collaborative exposition, covering topics from Hilbert schemes to crystalline cohomology precursors.^[24] Grothendieck's emphasis on "relative" objects—such as schemes over a base—resolved limitations in absolute varieties, enabling applications to number theory, including the study of arithmetic surfaces and the eventual proof of the Weil conjectures via étale cohomology.^[25] His methods demanded rethinking proofs in terms of universal properties, yielding tools like the Yoneda embedding for representability, which streamlined intersection theory and birational geometry.^[26] This paradigm shift elevated algebraic geometry to a cornerstone of modern mathematics, influencing fields from string theory to cryptography, as schemes offered a versatile language for gluing local data globally while preserving algebraic invariants.^[27] By 1966, Grothendieck received the Fields Medal for these foundational advances, recognizing how his work unified disparate threads into a coherent, general theory.^[25] Despite the abstraction's initial steep learning curve, it supplanted classical approaches, with subsequent texts like Hartshorne's Algebraic Geometry adapting its core concepts for broader accessibility.^[28]

Development of Cohomology Theories

Grothendieck's systematic approach to cohomology emerged from his reformulation of homological algebra, culminating in his 1957 paper "Sur quelques points d'algèbre homologique," published in the Tôhoku Mathematical Journal. In this work, he introduced abelian categories and derived functors, establishing that for any abelian category with enough injective objects—such as the category of sheaves of abelian groups on a topological space—cohomology groups could be defined as the right derived functors of the global sections functor.^[29] This framework generalized earlier ad hoc computations of sheaf cohomology, providing a universal tool for deriving cohomology in diverse settings beyond classical topology.^[30] Building on this foundation, Grothendieck applied derived functors to algebraic geometry in his 1958 paper "The cohomology theory of abstract algebraic varieties," presented at the International Congress of Mathematicians. Here, he proposed axioms for a "Weil cohomology theory" suitable for varieties over arbitrary fields, including finite fields, emphasizing properties like finite-dimensionality, Poincaré duality, and a Lefschetz trace formula to address André Weil's conjectures on zeta functions.^[31] Unlike classical de Rham or Betti cohomology, which fail over non-algebraically closed fields of positive characteristic, Grothendieck sought theories bridging algebraic and topological invariants, critiquing existing approaches like those of Serre for their limitations in handling non-constant coefficients or arithmetic varieties.^[19] To realize such a theory, Grothendieck introduced Grothendieck topologies (or sites) in the early 1960s, generalizing sheaf theory to allow cohomology computations on "non-Hausdorff" sites tailored to algebraic structures.^[32] His étale cohomology, developed primarily through the Séminaire de Géométrie Algébrique (SGA) volumes starting around 1960–1964, defined the étale site on schemes using étale morphisms—local isomorphisms in the sense of formally unramified and étale extensions—enabling l-adic sheaf cohomology that mimics Betti cohomology's properties for varieties over finite fields.^[19] This innovation, inspired partly by Serre's suggestions but formalized via toposes and higher derived functors, proved key to Deligne's 1974 resolution of the Weil conjectures, as the étale cohomology groups satisfy the required additivity and purity axioms.^[19] Grothendieck further extended this program with crystalline cohomology in the mid-1960s, targeting the p-torsion components elusive in étale theory for varieties in characteristic p. Defined using the crystalline site—comprising divided power thickenings of schemes—this theory provides a contravariant functor to vector spaces over Witt rings, compatible with de Rham cohomology in mixed characteristic and yielding Hodge decompositions absent in étale settings.^[33] These developments, detailed in SGA 4 and 6, unified disparate cohomology theories under abelian category machinery, prioritizing causal links between geometric structures and arithmetic invariants over ad hoc fixes, though later critiques noted the abstractness sometimes obscured concrete computations.^[19]

Advancements in Category Theory and Toposes

Grothendieck's foundational contributions to category theory emerged prominently in his 1957 paper "Sur quelques points d'algèbre homologique," published in the Tôhoku Mathematical Journal, where he introduced the concept of an abelian category possessing enough injective objects.^[34] This framework generalized the properties of modules over a ring to arbitrary abelian categories, including sheaves of abelian groups on topological spaces, enabling a unified treatment of homological algebra via derived functors and resolutions without reliance on specific concrete realizations.^[30] By axiomatizing exactness and exact sequences in this categorical setting, Grothendieck provided tools for computing cohomology in diverse contexts, such as sheaf cohomology, which paralleled Cartan's earlier work but extended it abstractly to facilitate applications in algebraic geometry.^[34] Building on this, Grothendieck integrated category-theoretic language deeply into algebraic geometry during the late 1950s and 1960s, particularly through his development of schemes and the Éléments de géométrie algébrique (EGA) series, co-authored with Jean Dieudonné starting in 1960.^[35] Here, he employed functors, adjointness, and limits/colimits to reformulate classical varieties in terms of relative schemes over base rings, emphasizing universal properties over coordinate-based descriptions and thereby revealing structural analogies across geometry, topology, and algebra.^[36] This categorical perspective not only streamlined proofs of existence via representable functors but also anticipated broader homotopical methods, influencing subsequent developments like derived categories, which Verdier formalized under Grothendieck's guidance around 1963.^[37] In the context of étale cohomology, pursued in the Séminaire de Géométrie Algébrique (SGA) volumes from 1960 onward, Grothendieck introduced Grothendieck topologies and toposes in the early 1960s to generalize sheaf theory beyond topological spaces.^[38] A Grothendieck topology on a category equips it with a notion of "covering families" satisfying axioms akin to open covers, allowing the definition of sheaves as functors satisfying descent conditions with respect to these covers; the resulting category of sheaves forms a topos, which inherits properties like having all finite limits, colimits, and subobject classifiers, generalizing the category of sets or sheaves on a space.^[36] This innovation, detailed in SGA 4 (circa 1964), enabled the construction of the étale topos on the étale site of a scheme, providing a cohomology theory analogous to singular cohomology but suited to algebraic varieties, with exactness properties ensuring computability via hypercoverings and Godement resolutions.^[32] Toposes thus served as "generalized spaces" for algebraic purposes, bridging point-set topology with abstract algebra and influencing logic through their internal language equivalence to geometric theories.^[38]

Ethical and Political Positions

Anti-War Activism and Protests

In 1967, Grothendieck traveled to North Vietnam for several weeks to protest the United States' military intervention in the conflict.^[4]^[39] There, he delivered lectures on advanced topics such as category theory at Hanoi University and in forested areas near the city, often under imminent threat of aerial bombardment by American forces.^[40] This direct action symbolized his pacifist stance and commitment to opposing imperialism, as he sought to support Vietnamese mathematicians amid the war's disruptions.^[41] Grothendieck's anti-war positions extended to criticism of Soviet military actions, reflecting a consistent opposition to expansionist policies on both sides of the Cold War divide.^[6] In France, his activism aligned with broader protests during the late 1960s, including the May 1968 uprisings, where he engaged with movements decrying militarism and foreign interventions.^[39] By the early 1970s, he co-founded the group Survivre with fellow mathematicians Claude Chevalley and Pierre Samuel, which organized efforts against war, imperialism, and environmental degradation linked to military activities.^[42] These initiatives included public campaigns for disarmament and against nuclear proliferation, though they achieved limited institutional impact.^[6]

Critiques of Scientific Institutions and Militarization

In 1970, Grothendieck resigned from the Institut des Hautes Études Scientifiques (IHÉS) upon discovering that a portion of its budget—never exceeding five percent—derived indirectly from the French Ministry of Defense, which he viewed as enabling militarization of scientific endeavors.^[42]^[43] He had previously attempted to persuade IHÉS director Léon Motchane and colleagues to reject such funds, arguing that any acceptance implicated the institution in military applications, but faced resistance, prompting his abrupt departure after over a decade of affiliation.^[42] This act symbolized his broader rejection of scientific bodies that tolerated ties to defense-related entities, such as the Commissariat à l'Énergie Atomique, which supported nuclear programs with dual military and civilian aims.^[43] Grothendieck articulated these concerns in speeches, including one on June 26, 1970, at the Orsay Faculty of Sciences, and another on July 8, 1970, at the University of Montreal, later compiled as "The Responsibility of the Scientist Today."^[44] Therein, he condemned scientists' widespread acceptance of military subventions—such as short-term contracts valued at around $2,000 in the U.S.—as moral abdication and direct complicity in warfare, equating it to bolstering destructive capabilities under the guise of "pure" research.^[44] He emphasized that scientists, as creators of enabling technologies like nuclear weapons, held disproportionate responsibility for existential threats and urged total non-cooperation, dismissing justifications for fund diversion as hypocritical rationalizations that sustained military power.^[44] These views extended to institutional critique in his involvement with the Survivre et Vivre group, co-founded on July 27, 1970, in Montreal, which advocated pacifism, ecology, and opposition to militarism by decoupling science from state aggression. In later reflections, such as Récoltes et Semailles (written 1983–1986), Grothendieck decried the fusion of scientific progress with atomic militarism—evoking Hiroshima—and portrayed Western scientific communities as passively endorsing "barbarian" apparatuses through funding dependencies, prioritizing institutional survival over ethical imperatives.^[45] His stance highlighted a causal link between unexamined financial ties and amplified global risks, insisting on personal accountability amid systemic incentives for collaboration.^[44]

Declining Engagement and Reclusion

Resignation from IHÉS and Withdrawal

Alexander Grothendieck resigned from his position as permanent professor at the Institut des Hautes Études Scientifiques (IHÉS) via a letter dated May 25, 1970.^[6] The decision stemmed from his discovery that IHÉS received indirect funding from the French Ministry of Defense, a revelation that conflicted with his deepening opposition to militarism and the perceived complicity of scientific institutions in military applications.^[5]^[19] Grothendieck had urged IHÉS director Léon Motchane to reject such funding, but Motchane, who prioritized separating scientific pursuits from political influences, refused, leading to irreconcilable tensions.^[5] This principled stand aligned with Grothendieck's broader ethical critiques of science's alignment with state power, though some contemporaries viewed his reaction as overly absolutist given the funding's minor scale.^[46] Following the resignation, Grothendieck progressively withdrew from active engagement in the mathematical community, ceasing collaborations and mainstream publications by the early 1970s.^[6] While he briefly pursued ecological and anti-militarization activism through groups like Survivre, his involvement diminished, marking a shift toward personal reflection and disengagement from institutional academia.^[47] This retreat was not abrupt but evolved from disillusionment with professional norms and a quest for deeper existential inquiry, as evidenced in his later autobiographical writings critiquing the mathematical establishment's priorities.^[48] By refusing awards like the Fields Medal's monetary prize in 1966—donating it to dissident causes—and later shunning recognition, Grothendieck exemplified a deliberate severance from the accolades and networks of elite mathematics.^[6] His withdrawal preserved his intellectual independence amid growing personal isolation, influencing perceptions of him as both a moral exemplar and an enigmatic figure detached from conventional scholarly life.^[39]

1980s Manuscripts and Personal Reflections

In the early 1980s, following his withdrawal from institutional mathematics, Grothendieck composed "La Longue Marche à travers la Théorie de Galois" between January and June 1981, an extensive manuscript of approximately 1600 pages that reinterprets classical Galois theory through modern lenses such as étale cohomology, anabelian geometry, and the role of fundamental groups in reconstructing algebraic varieties.^[49] This work, circulated privately among a few colleagues, emphasizes "descent" processes and the interplay between arithmetic and geometry, foreshadowing ideas in his later "Esquisse d'un Programme" of 1984, while critiquing overly rigid foundational approaches in favor of intuitive, structural insights.^[50] The manuscript remained unpublished during his lifetime, with drafts preserved in archives, reflecting his solitary pursuit of unifying themes across number theory and topology without reliance on contemporary collaborations.^[51] From June 1983 to around 1986, Grothendieck produced "Récoltes et Semailles," a sprawling autobiographical reflection exceeding 900 pages, intended as the first installment in a planned series titled "Réflexions Mathématiques" (Mathematical Reflections).^[52] Divided into preludes and thematic sections, it chronicles his career trajectory, mathematical intuitions—described via the metaphor of a "rising sea" submerging obstacles to reveal underlying structures—and profound disillusionment with the profession's interpersonal dynamics, including perceived betrayals by figures like Pierre Deligne, whom he accused of exploiting his ideas without adequate reciprocity.^[53] Grothendieck attributes his 1970 resignation from the Institut des Hautes Études Scientifiques (IHÉS) to irreconcilable ethical conflicts, portraying mathematics not as a competitive enterprise but as a vocation demanding purity and communal harmony, which he found absent amid institutional politics and militarized funding.^[47] These manuscripts interweave technical exposition with personal testimony, revealing Grothendieck's evolving mysticism, where mathematical discovery parallels spiritual awakening, and critique extends to broader societal ills like environmental degradation and authoritarianism, echoing his earlier activism.^[45] Distributed selectively to acquaintances, they underscore his isolation, as he rejected publication offers and academic honors, prioritizing unfiltered expression over dissemination.^[54] Partial editions appeared posthumously, with a truncated French version released by Gallimard in 2022, though full manuscripts are accessible via digital archives, affirming their status as raw, unpolished testaments to his intellectual odyssey.^[55]

Final Years, Seclusion, and Death

In the late 1980s, Grothendieck declined the Crafoord Prize, a prestigious award accompanied by significant monetary recognition, reflecting his growing disengagement from institutional honors.^[6] By August 1991, he abruptly left his residence and retreated to an undisclosed location, severing nearly all contact with others and embracing a hermitic existence.^[6] He eventually settled in a remote, unpowered house without running water in Lasserre, a village in the Ariège department of the French Pyrenees, where he resided alone until his death.^[6]^[56] During his two decades in Lasserre, Grothendieck sustained himself through simple means, including gardening and foraging, while producing thousands of pages of writings that delved into physics, philosophical inquiries on determinism and free will, metaphysics, the nature of evil, and structural aspects of the psyche.^[6]^[56] These manuscripts, often accompanied by topological diagrams and numerological elements, were not published during his lifetime and reflected a shift toward esoteric and interdisciplinary themes rather than conventional mathematics.^[56] He maintained extreme isolation, politely rebuffing most visitors and exhibiting signs of paranoia, such as viewing some as adversarial figures or fearing external threats, which limited interactions even with locals and estranged family.^[8]^[56] In his final months, Grothendieck, afflicted by profound deafness and near-blindness, requested a firearm from a neighbor amid declining health, prompting rare outreach to his children after years of separation.^[56] He died on November 13, 2014, at the age of 86, in a hospital in Saint-Girons, Ariège, reportedly from exhaustion without specified underlying causes.^[6]^[56] His body was buried in the Lasserre churchyard under a simple sandstone slab.^[56]

Personal Life and Identity

Family Dynamics and Relationships

Alexander Grothendieck's parents, Alexander "Sascha" Schapiro (1890–1942) and Johanna "Hanka" Grothendieck (1900–1957), shared a background steeped in political radicalism. Schapiro, a Russian Jew and anarchist who had participated in revolutionary activities including the 1905 Russian uprising and the Spanish Civil War, met Hanka, a German journalist from a Protestant bourgeois family in Hamburg with socialist leanings, in Berlin in the late 1920s.^[6]^[8]^[57] Their union produced Grothendieck in 1928, but the couple separated amid the escalating political turmoil of the 1930s, with Schapiro pursuing anarchist causes and Hanka managing practical survival.^[8] In 1939, Hanka fled with young Grothendieck to France to escape Nazi persecution, while Schapiro, who had earlier reunited briefly with them, was interned by the Vichy regime and deported to Auschwitz, where he was killed in 1942.^[58]^[43] Grothendieck did not learn of his father's fate until 1946, after the war, and subsequently idealized Schapiro as a moral exemplar despite their limited time together, which profoundly shaped his later ethical and anti-militaristic views.^[8] In contrast, Grothendieck endured separations during the war, living in French internment camps with his mother before being placed with relatives; post-liberation in 1945, he reunited with Hanka and maintained a close attachment to her until her death in 1957.^[59] This dynamic—distant paternal influence versus maternal proximity—underscored a childhood marked by instability and ideological inheritance rather than conventional family stability.^[6] In adulthood, Grothendieck's personal relationships mirrored elements of detachment. He married Mireille Dufour around the late 1950s or early 1960s, and they had three children: Johanna, Mathieu, and Alexandre.^[6]^[60] The marriage dissolved amid Grothendieck's shifting lifestyle, including the establishment of communes in the 1970s following his withdrawal from institutional mathematics.^[43]^[59] He fathered additional children outside this union, including a son John from an early relationship that ended shortly after the child's birth, resulting in minimal ongoing contact, and another son with Justine Skalba.^[8]^[43] Grothendieck's interactions with his children evolved unevenly, influenced by his increasing reclusion and spiritual pursuits, though specific details remain limited due to his private nature.^[61]

Citizenship, Heritage, and Self-Perception

Alexander Grothendieck was born on 28 March 1928 in Berlin, Germany, to parents with revolutionary backgrounds. His father, Alexander "Sascha" Schapiro (1890–1942), was a Russian Jew from Novozybkov who participated in the 1905 Revolution, lost his left arm in the fighting, and later perished in Auschwitz after deportation by the Vichy regime.^[6] His mother, Johanna "Hanka" Grothendieck (1900–1957), hailed from Hamburg and engaged in left-wing political groups during the 1920s.^[6] The parents, who were not formally married at his birth, shared anarchist convictions and supported pacifist causes, including involvement in the Spanish Civil War; Grothendieck adopted his mother's surname.^[43] His paternal heritage was Jewish, though his father rejected religious orthodoxy in favor of atheism, while his mother's German Protestant roots contributed to a secular family ethos devoid of strong ethnic or religious identifications.^[62] Fleeing Nazi persecution, Grothendieck arrived in France as a refugee in 1939, where wartime internment of his parents under the 1938 "undesirables" law rendered him effectively stateless; German records of his birth nationality were destroyed in 1945.^[6]^[63] He relied on a Nansen passport, a League of Nations document for refugees, for international travel and long avoided French naturalization due to mandatory military service requirements, particularly amid opposition to the Algerian War.^[42] Accounts differ on the timing of his eventual French citizenship, with some indicating naturalization in 1971 and others in the late 1980s after prolonged statelessness.^[64]^[65] Grothendieck perceived himself as a "citizen of the world" unbound by national loyalties, reflecting his parents' anarchist heritage and his own pacifism; in 1966, he formally requested United Nations citizenship and declined to attend the Fields Medal ceremony in Moscow as a protest against militarism.^[6] He embraced a universalist identity over ethnic or patriotic ties, viewing statelessness not as a burden but as alignment with anti-authoritarian principles, and maintained an atheistic stance uninfluenced by Judaism or Protestantism despite his mixed ancestry.^[63]^[66] This outlook underpinned his rejection of institutional nationalism throughout his career.^[42]

Enduring Influence and Reception

Grothendieck's introduction of schemes in the 1960s fundamentally reshaped algebraic geometry by generalizing classical varieties to affine schemes associated with commutative rings, enabling a uniform treatment of geometric objects over arbitrary base rings rather than restricting to fields. This framework incorporated Spec(R) as the geometric counterpart to the ring R, allowing arithmetic geometry to merge seamlessly with classical algebraic geometry and resolving longstanding issues in handling singular or non-reduced structures.^[19]^[67] A pivotal transformation arose from Grothendieck's generalization of the Riemann-Roch theorem, culminating in the Grothendieck-Riemann-Roch theorem announced in 1958, which equates the pushforward of a class in the Grothendieck group of coherent sheaves under a proper morphism with the Chern character composed with the Todd class of the target. This result, proven using K-theory and intersection theory, provided a universal tool for computing Euler characteristics of vector bundles on schemes, extending beyond smooth projective varieties and influencing subsequent developments in characteristic classes.^[68]^[69] In parallel, Grothendieck's development of topos theory during the 1960s, motivated by étale cohomology, abstracted sheaf cohomology to Grothendieck topoi defined via sites—a category equipped with a Grothendieck topology for coverings—transforming geometric intuition into categorical terms where topoi behave like generalized spaces supporting internal logic and cohomology. This enabled the construction of new cohomology theories, such as l-adic étale cohomology, which proved the Riemann hypothesis for varieties over finite fields as part of the Weil conjectures resolved by Deligne in 1974, thereby bridging algebraic geometry with number theory through functorial transformations.^[38]^[70] These innovations extended to related fields like homological algebra, where Grothendieck's emphasis on derived categories and six functor formalisms provided tools for exact computations of sheaf cohomology, influencing motivic cohomology and the standard conjectures on algebraic cycles. His relative point of view, treating morphisms between schemes as fundamental, shifted geometric reasoning from absolute properties to transformations under base change and fiber products, fostering applications in deformation theory and moduli problems.^[19]^[4]

Broader Impacts on Mathematics and Beyond

Grothendieck's étale cohomology, developed around 1958 through the introduction of the étale topology on schemes, furnished the algebraic analogues of singular cohomology necessary for Pierre Deligne's 1974 proof of the Weil conjectures, thereby forging deep connections between algebraic geometry and Diophantine problems in number theory.^[71]^[72] This framework enabled the computation of zeta functions for varieties over finite fields and extended topological methods to arithmetic settings, influencing subsequent advances in the Langlands program.^[73] His vision of motives, proposed in the 1960s as universal objects underlying diverse cohomology theories, sought to distill the "heart" of arithmetic geometry by classifying varieties up to numerical equivalence, posing a central challenge that persists in efforts to realize a motivic cohomology.^[74]^[75] This program has spurred research into standard conjectures on algebraic cycles and their role in unifying Hodge, étale, and l-adic cohomologies, with implications for understanding zeta functions and L-functions beyond classical cases.^[76] Grothendieck's axiomatic recasting of homological algebra in 1957, via derived categories and abstract abelian categories, propelled category theory from a linguistic tool to an autonomous discipline, permeating homological methods across topology, algebra, and analysis.^[77] Similarly, Grothendieck topoi, arising from sheaf theory on sites to resolve issues in algebraic geometry like the Weil conjectures, generalize spaces in ways that encode local-to-global principles.^[70] Beyond core mathematics, topos theory provides a categorical semantics for intuitionistic logic through internal Heyting algebras and subobject classifiers, offering an alternative to classical set theory for foundational inquiries.^[77]^[70] In theoretical computer science, it underpins type theory and domain theory for programming languages, facilitating proofs of correctness via categorical models.^[77] Applications extend tentatively to physics, as in the Bohr topos for contextual quantum mechanics, where topoi model non-Boolean event structures without collapsing wave functions.^[70]

Recent Reassessments and Publications

In 2022, Éditions Gallimard released the complete edition of Grothendieck's Récoltes et semailles, a multi-volume autobiographical manuscript composed between February 1984 and May 1986, comprising over 2,800 pages of reflections on his mathematical career, philosophical insights, and personal motivations.^[78] ^[79] This publication, previously circulated in limited photocopied form among colleagues, details Grothendieck's self-described "great ideas" in algebraic geometry, such as schemes, toposes, and étale cohomology, while critiquing institutional dynamics in mathematics.^[79] Posthumously, initiatives to disseminate Grothendieck's unpublished writings gained momentum. In 2015, the University of Montpellier acquired roughly 20,000 pages of his personal notes, including mathematical drafts from 1949 to 1991 and later philosophical texts, with portions digitized and accessible online since May 2017.^[80] ^[81] The Centro Studi Grothendieckiani, established to honor his legacy, coordinates scholarly editions of these materials, encompassing mathematical reflections, ethical essays, and literary pieces from his reclusive period.^[82] ^[83] These efforts have spurred reassessments of Grothendieck's foundational contributions. For instance, analyses of his functional analysis papers from the 1950s highlight their role in advancing nuclear spaces and tensor products, influencing modern operator theory.^[84] Edited volumes, such as those examining his topos theory and philosophical implications, underscore how his abstract frameworks continue to underpin advancements in geometry, logic, and category theory, with ongoing seminars and journals like the Quaderni del Centro di Studi Grothendieckiani facilitating deeper exploration since 2016.^[85] ^[86] The 2022 release of Récoltes et semailles has particularly prompted reevaluations of his "rising sea" metaphor for conceptual unification, affirming its causal efficacy in resolving longstanding problems in algebraic varieties and motives.^[79]

Key Publications and Archives

Grothendieck's foundational contributions to homological algebra appeared in his 1957 paper "Sur quelques points d'algèbre homologique," published in the Tôhoku Mathematical Journal, which introduced derived categories and six functor formalisms for sheaf cohomology. His systematic reconstruction of algebraic geometry is detailed in the Éléments de géométrie algébrique (EGA), co-authored with Jean Dieudonné: volume I (Le langage des schémas) in 1960, volume II (Étude globale élémentaire de quelques classes de morphismes) in 1961, volume III (Étude cohomologique des faisceaux cohérents) in 1961, and volume IV in four parts from 1964 to 1967, all issued through Publications Mathématiques de l'IHÉS.^[87] The Séminaire de Géométrie Algébrique (SGA) series, directed by Grothendieck at IHÉS from 1960 to 1969, comprises seven main volumes plus exposés, covering topics from étale cohomology (SGA 4, 1963–1965) to motives and crystalline cohomology (SGA 5 and 6, 1966–1968), with later volumes like SGA 7 on Néron models (1970).^[88] These seminar proceedings, edited post-facto, formalized advanced tools in scheme theory and l-adic cohomology. Later published works include Esquisse d'un programme (1984), outlining anabelian geometry and the "dessins d'enfants" approach to Galois representations, first circulated privately and reprinted in Geometric Galois Actions (1997).^[89] Récoltes et semailles (1985–1986), a reflective manuscript on mathematical practice and personal experience, was published in two volumes by Gallimard in 2022.^[54] Post-resignation from IHÉS in 1970, Grothendieck generated over 60,000 pages of unpublished manuscripts, known as the Lasserre collection (1992–1999), encompassing revisions of algebraic geometry foundations, Galois theory extensions, and philosophical notes.^[90] These were deposited at the University of Montpellier around 2010, with approximately 28,000 pages digitized and partially accessible online since 2017, including typed notes and letters.^[81]^[91] In September 2023, a portion of these unpublished works was transferred to the Bibliothèque nationale de France (BnF) for preservation and scholarly access.^[92] Earlier archives, such as seminar drafts, remain at IHÉS.^[81]

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[82]
Some of Grothendieck's archives published online by the University ...
May 23, 2017 · Alexandre Grothendieck is considered one of the greatest mathematicians of the 20th century. He revolutionised the foundations and the ...
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Unpublished works - CSG Istituto Grothendieck
This self-imposed isolation bequeathed us an extensive collection of intermittent, both lengthy and concise, mathematical reflections and literary meditations.
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CSG Istituto Grothendieck: Home
Editorial program. Coordination of editions of unpublished Grothendieck manuscripts, either mathematical, literary, philosophical, ethical or sociological.Missing: 2020-2025 | Show results with:2020-2025
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https://link.springer.com/book/10.1007/978-3-031-68934-5
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The Mathematical and Philosophical Legacy of Alexander ...
This edited volume utilizes a multidisciplinary approach to examine Alexander Grothendieck's lasting influence on mathematics, philosophy, and logic.Missing: Istituto | Show results with:Istituto
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Have Grothendieck's notes in Montpellier already been investigated?
Feb 11, 2016 · You could keep an eye on the Quaderni del Centro di Studi Grothendieckiani, a biennial refereed journal dedicated to exploring Alexander ...
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Éléments de géométrie algébrique : II. Étude globale ... - Numdam
Grothendieck, Alexander. Publications Mathématiques de l'IHÉS, Tome 8 (1961), pp. 5-222. partie précédente Éléments de géométrie algébrique : I. Le langage ...
[89]
SGA
Archive of scans that we created of SGA, etc. Spanish site with huge amount of work by Grothendieck. Click here for a PDF version of the SGA scans.
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Esquisse d'un Programme - Geometric Galois Actions
Apr 5, 2013 · Esquisse d'un Programme. Published online by Cambridge University Press: 05 April 2013. By. Alexandre Grothendieck. Edited by. Leila Schneps and.
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[LUM#1] In search of Alexandre Grothendieck
Jan 19, 2016 · Drawn up between 1992 and 1999 in Grothendieck's final residence, the "Lasserre collection" contains his last papers: 60,000 pages, including ...Missing: Alexander | Show results with:Alexander<|separator|>
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28,000 pages by Alexandre Grothendieck - Université de Montpellier
May 10, 2017 · The Grothendieck collection includes unpublished manuscripts of major twentieth-century mathematical theories, particularly algebraic geometry.
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Handover of unpublished manuscripts of A. Grothendieck to the BnF
Sep 28, 2023 · On 28 September 2023, a ceremony was held in Paris, at the premises of the Bibliothèque nationale de France (BnF), at which Alexander ...