154-447: Alexander Grothendieck , later Alexandre Grothendieck in French ( / ˈ ɡ r oʊ t ən d iː k / ; German: [ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk] ; French: [ɡʁɔtɛndik] ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry . His research extended the scope of
308-665: A Nansen passport , given that he had refused to take French nationality (as that would have entailed military service against his convictions). He stayed in São Paulo (apart from a lengthy visit in France from October 1953 to March 1954) until the end of 1954. His published work from the time spent in Brazil is still in the theory of topological vector spaces; it is there that he completed his last major work on that topic (on "metric" theory of Banach spaces ). Grothendieck moved to Lawrence, Kansas at
462-637: A Neutrality Pact in April 1941, and Japan adopted the doctrine of Nanshin-ron , promoted by the Navy, which took its focus southward and eventually led to war with the United States and the Western Allies. In Europe, Germany and Italy were becoming more aggressive. In March 1938, Germany annexed Austria , again provoking little response from other European powers. Encouraged, Hitler began pressing German claims on
616-492: A Protestant German family in Hamburg and worked as a journalist. As teenagers, both of his parents had broken away from their early backgrounds. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and initially, his birth name was recorded as "Alexander Raddatz." That marriage was dissolved in 1929 and Schapiro acknowledged his paternity, but never married Hanka Grothendieck. Grothendieck had
770-564: A cautious French probe into the Saarland . The Western Allies also began a naval blockade of Germany , which aimed to damage the country's economy and war effort. Germany responded by ordering U-boat warfare against Allied merchant and warships, which would later escalate into the Battle of the Atlantic . On 8 September, German troops reached the suburbs of Warsaw . The Polish counter-offensive to
924-413: A commune in the early 1970s. 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
1078-453: A 600-page manuscript entitled Pursuing Stacks . It began 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
1232-479: 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 recently had been generalized to any dimension by Hirzebruch . The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at
1386-428: A connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers . Weil had realized that to prove such a connection, one needed a new cohomology theory, but neither he nor any other expert saw how to accomplish this until such
1540-501: A deep compassion for the poor and the downtrodden. As Cartier puts it, Grothendieck came to find Bures-sur-Yvette as " une cage dorée " ("a gilded cage"). While Grothendieck was at the IHÉS, opposition to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a mandarin of the scientific world. In addition, after several years at
1694-471: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include
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#17328583477581848-417: 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 his relative point of view (pairs of varieties related by a morphism ), allowing a broad generalization of many classical theorems. The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on
2002-520: A list of 14 open questions, relevant for locally convex spaces . Grothendieck introduced new mathematical methods that enabled him to solve all of these problems within a few months. In Nancy, he wrote his dissertation under those two professors on functional analysis , from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. In 1953 he moved to the University of São Paulo in Brazil, where he immigrated by means of
2156-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while
2310-589: A maternal sibling, his half sister Maidi. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism . His mother followed soon thereafter. Grothendieck was left in the care of Wilhelm Heydorn, a Lutheran pastor and teacher in Hamburg . According to Winfried Scharlau , during this time, his parents took part in the Spanish Civil War as non-combatant auxiliaries. However, others state that Schapiro fought in
2464-417: A new address that he did not share with his previous contacts in the mathematical community. Very few people visited him afterward. Local villagers helped sustain him with a more varied diet after he tried to live on a staple of dandelion soup. At some point, Leila Schneps and Pierre Lochak [ fr ] located him, then carried on a brief correspondence. Thus they became among "the last members of
2618-766: A political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages
2772-485: A political group entitled Survivre —the name later changed to Survivre et vivre . The group published a bulletin and was dedicated to antimilitary and ecological issues. It also developed strong criticism of the indiscriminate use of science and technology. Grothendieck devoted the next three years to this group and served as the main editor of its bulletin. Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly ended when he left
2926-514: A pretext to initiate the invasion. The first German attack of the war came against the Polish defences at Westerplatte . The United Kingdom responded with an ultimatum for Germany to cease military operations, and on 3 September, after the ultimatum was ignored, Britain and France declared war on Germany. During the Phoney War period, the alliance provided no direct military support to Poland, outside of
3080-765: A pretext to worsen relations. On 29 August, Hitler demanded that a Polish plenipotentiary immediately travel to Berlin to negotiate the handover of Danzig , and to allow a plebiscite in the Polish Corridor in which the German minority would vote on secession. The Poles refused to comply with the German demands, and on the night of 30–31 August in a confrontational meeting with the British ambassador Nevile Henderson , Ribbentrop declared that Germany considered its claims rejected. On 1 September 1939, Germany invaded Poland after having staged several false flag border incidents as
3234-406: A refugee. Records of his nationality were destroyed in the fall of Nazi Germany in 1945 and he did not apply for French citizenship after the war. Thus, he became a stateless person for at least the majority of his working life and he traveled on a Nansen passport . Part of his reluctance to hold French nationality is attributed to not wishing to serve in the French military, particularly due to
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#17328583477583388-597: A relation between Grothendieck inequality and the Einstein–Podolsky–Rosen paradox in quantum physics . In 1958, Grothendieck was installed at the Institut des hautes études scientifiques (IHÉS), a new privately funded research institute that, in effect, had been created for Jean Dieudonné and Grothendieck. Grothendieck attracted attention by an intense and highly productive activity of seminars there ( de facto working groups drafting into foundational work some of
3542-469: A secret protocol that defined German and Soviet "spheres of influence" (western Poland and Lithuania for Germany; eastern Poland , Finland, Estonia , Latvia and Bessarabia for the Soviet Union), and raised the question of continuing Polish independence. The pact neutralised the possibility of Soviet opposition to a campaign against Poland and assured that Germany would not have to face the prospect of
3696-743: A similar pact and rejected ceding part of its territory to the Soviet Union. The Soviet Union invaded Finland in November 1939, and was subsequently expelled from the League of Nations for this crime of aggression. Despite overwhelming numerical superiority, Soviet military success during the Winter War was modest, and the Finno-Soviet war ended in March 1940 with some Finnish concessions of territory . In June 1940,
3850-449: A theory was expressed by Grothendieck. Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of the earliest known mathematicians was Thales of Miletus ( c. 624 – c. 546 BC ); he has been hailed as
4004-519: A two-front war, as it had in World War I. Immediately afterwards, Hitler ordered the attack to proceed on 26 August, but upon hearing that the United Kingdom had concluded a formal mutual assistance pact with Poland and that Italy would maintain neutrality, he decided to delay it. In response to British requests for direct negotiations to avoid war, Germany made demands on Poland, which served as
4158-466: A website devoted to his work as "an abomination". His dictate may have been reversed in 2010. In September 2014, almost totally deaf and blind, he asked a neighbour to buy him a revolver so he could kill himself.On 13 November 2014, aged 86, Grothendieck died in the hospital of Saint-Lizier or Saint-Girons, Ariège . Grothendieck was born in Weimar Germany . In 1938, aged ten, he moved to France as
4312-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics
4466-518: Is Grothendieck's account of how his consideration of the source of dreams led him to conclude that a deity exists. As part of the notes to this manuscript, Grothendieck described the life and the work of 18 "mutants", people whom he admired as visionaries far ahead of their time and heralding a new age. The only mathematician on his list was Bernhard Riemann . Influenced by the Catholic mystic Marthe Robin who
4620-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics
4774-546: Is also available on the Internet. An English translation by Leila Schneps will be published by MIT Press in 2025. A partial English translation can be found on the Internet. A Japanese translation of the whole book in four volumes was completed by Tsuji Yuichi (1938–2002), a friend of Grothendieck from the Survivre period. The first three volumes (corresponding to Parts 0 to III of the book) were published between 1989 and 1993, while
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4928-489: 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 approximately 2000 pages, further developed the homotopical ideas begun in Pursuing Stacks . Much of this work anticipated the subsequent development during the mid-1990s of the motivic homotopy theory of Fabien Morel and Vladimir Voevodsky . In 1984, Grothendieck wrote
5082-527: Is collected in the monumental, yet incomplete, Éléments de géométrie algébrique ( EGA ) and Séminaire de géométrie algébrique ( SGA ). The collection Fondements de la Géometrie Algébrique ( FGA ), which gathers together talks given in the Séminaire Bourbaki , also contains important material. Grothendieck's work includes the invention of the étale and l-adic cohomology theories, which explain an observation made by André Weil that argued for
5236-579: Is described as the "Golden Age" period at the IHÉS established several unifying themes in algebraic geometry , number theory , topology , category theory , and complex analysis . His first (pre-IHÉS) discovery in algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem , a generalisation of the Hirzebruch–Riemann–Roch theorem proved algebraically; in this context he also introduced K-theory . Then, following
5390-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing
5544-689: The Algerian War (1954–62). He eventually applied for French citizenship in the early 1980s, after he was well past the age that would have required him to do military service. Grothendieck was very close to his mother, to whom he dedicated his dissertation. She died in 1957 from tuberculosis that she contracted in camps for displaced persons . He had five children: a son with his landlady during his time in Nancy; three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour; and one child with Justine Skalba, with whom he lived in
5698-588: The Anglo-German Naval Agreement and the German–Polish declaration of non-aggression . The situation became a crisis in late August as German troops continued to mobilise against the Polish border. On 23 August the Soviet Union signed a non-aggression pact with Germany, after tripartite negotiations for a military alliance between France, the United Kingdom, and Soviet Union had stalled. This pact had
5852-662: The British Empire , with fighting in the Balkans , Mediterranean, and Middle East , the aerial Battle of Britain and the Blitz , and naval Battle of the Atlantic . Through a series of campaigns and treaties, Germany took control of much of continental Europe and formed the Axis alliance with Italy , Japan , and other countries. In June 1941, Germany led the European Axis in an invasion of
6006-470: The Crafoord Prize with an open letter to the media. He wrote that he and other established mathematicians had no need for additional financial support and criticized what he saw as the declining ethics of the scientific community that was characterized by outright scientific theft that he believed had become commonplace and tolerated. The letter also expressed his belief that totally unforeseen events before
6160-566: The German Revolution of 1918–1919 , and a democratic government, later known as the Weimar Republic , was created. The interwar period saw strife between supporters of the new republic and hardline opponents on both the political right and left. Italy, as an Entente ally, had made some post-war territorial gains; however, Italian nationalists were angered that the promises made by the United Kingdom and France to secure Italian entrance into
6314-814: The German invasion of Poland and the United Kingdom and France 's declaration of war on Germany two days later on 3 September 1939. Dates for the beginning of the Pacific War include the start of the Second Sino-Japanese War on 7 July 1937, or the earlier Japanese invasion of Manchuria , on 19 September 1931. Others follow the British historian A. J. P. Taylor , who stated that the Sino-Japanese War and war in Europe and its colonies occurred simultaneously, and
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6468-567: The Italian mainland , and Allied offensives in the Pacific—cost the Axis powers their initiative and forced them into strategic retreat on all fronts. In 1944, the Western Allies invaded German-occupied France at Normandy , while the Soviet Union regained its territorial losses and pushed Germany and its allies westward. At the same time, Japan suffered reversals in mainland Asia, while the Allies crippled
6622-537: The Japanese Navy and captured key islands . The war in Europe concluded with the liberation of German-occupied territories ; the invasion of Germany by the Western Allies and the Soviet Union, culminating in the fall of Berlin to Soviet troops; Hitler's suicide ; and the German unconditional surrender on 8 May 1945 . Following the refusal of Japan to surrender on the terms of the Potsdam Declaration ,
6776-554: 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 Le Chambon attended Collège Cévenol, and it was at this school that Grothendieck apparently first became fascinated with mathematics. In 1990, for risking their lives to rescue Jews, the entire village was recognized as " Righteous Among
6930-682: The League of Nations as a force to preserve peace. Both Italy and Ethiopia were member nations, but the League did little when the former clearly violated Article X of the League's Covenant . The United Kingdom and France supported imposing sanctions on Italy for the invasion, but the sanctions were not fully enforced and failed to end the Italian invasion. Italy subsequently dropped its objections to Germany's goal of absorbing Austria . When civil war broke out in Spain, Hitler and Mussolini lent military support to
7084-693: The Nationalist rebels , led by General Francisco Franco . Italy supported the Nationalists to a greater extent than the Nazis: Mussolini sent more than 70,000 ground troops, 6,000 aviation personnel, and 720 aircraft to Spain. The Soviet Union supported the existing government of the Spanish Republic . More than 30,000 foreign volunteers, known as the International Brigades , also fought against
7238-677: The Rome–Berlin Axis . A month later, Germany and Japan signed the Anti-Comintern Pact , which Italy joined the following year. The Kuomintang (KMT) party in China launched a unification campaign against regional warlords and nominally unified China in the mid-1920s, but was soon embroiled in a civil war against its former Chinese Communist Party (CCP) allies and new regional warlords . In 1931, an increasingly militaristic Empire of Japan , which had long sought influence in China as
7392-656: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of
7546-581: The Soviet–Japanese Neutrality Pact in April 1941. By contrast, the Germans were steadily making preparations for an attack on the Soviet Union, massing forces on the Soviet border. Hitler believed that the United Kingdom's refusal to end the war was based on the hope that the United States and the Soviet Union would enter the war against Germany sooner or later. On 31 July 1940, Hitler decided that
7700-580: The Stresa Front in April 1935 in order to contain Germany, a key step towards military globalisation ; however, that June, the United Kingdom made an independent naval agreement with Germany, easing prior restrictions. The Soviet Union, concerned by Germany's goals of capturing vast areas of Eastern Europe , drafted a treaty of mutual assistance with France. Before taking effect, though, the Franco-Soviet pact
7854-689: The Sudetenland , an area of Czechoslovakia with a predominantly ethnic German population. Soon the United Kingdom and France followed the appeasement policy of British Prime Minister Neville Chamberlain and conceded this territory to Germany in the Munich Agreement , which was made against the wishes of the Czechoslovak government, in exchange for a promise of no further territorial demands. Soon afterwards, Germany and Italy forced Czechoslovakia to cede additional territory to Hungary, and Poland annexed
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#17328583477588008-682: The Tanggu Truce was signed in 1933. Thereafter, Chinese volunteer forces continued the resistance to Japanese aggression in Manchuria , and Chahar and Suiyuan . After the 1936 Xi'an Incident , the Kuomintang and CCP forces agreed on a ceasefire to present a united front to oppose Japan. The Second Italo-Ethiopian War was a brief colonial war that began in October 1935 and ended in May 1936. The war began with
8162-466: The Trans-Olza region of Czechoslovakia. Although all of Germany's stated demands had been satisfied by the agreement, privately Hitler was furious that British interference had prevented him from seizing all of Czechoslovakia in one operation. In subsequent speeches Hitler attacked British and Jewish "war-mongers" and in January 1939 secretly ordered a major build-up of the German navy to challenge British naval supremacy. In March 1939, Germany invaded
8316-506: The Treaty of Craiova . The loss of one-third of Romania's 1939 territory caused a coup against King Carol II, turning Romania into a fascist dictatorship under Marshal Ion Antonescu , with a course set towards the Axis in the hopes of a German guarantee. Meanwhile, German-Soviet political relations and economic co-operation gradually stalled, and both states began preparations for war. In April 1940, Germany invaded Denmark and Norway to protect shipments of iron ore from Sweden , which
8470-401: The United Kingdom and France to declare war on Germany. Poland was divided between Germany and the Soviet Union under the Molotov–Ribbentrop Pact , in which they had agreed on " spheres of influence " in Eastern Europe. In 1940, the Soviets annexed the Baltic states and parts of Finland and Romania . After the fall of France in June 1940, the war continued mainly between Germany and
8624-426: The Wehrmacht rapidly advanced to the Channel and cut off the Allied forces in Belgium, trapping the bulk of the Allied armies in a cauldron on the Franco-Belgian border near Lille. The United Kingdom was able to evacuate a significant number of Allied troops from the continent by early June, although they had to abandon almost all their equipment. On 10 June, Italy invaded France , declaring war on both France and
8778-451: The aftermath of World War I and the rise of fascism in Europe and militarism in Japan . Key events leading up to the war included Japan's invasion of Manchuria , the Spanish Civil War , the outbreak of the Second Sino-Japanese War , and Germany's annexations of Austria and the Sudetenland . World War II is generally considered to have begun on 1 September 1939, when Nazi Germany , under Adolf Hitler , invaded Poland , prompting
8932-448: The derived category theory), and Pierre Deligne . Collaborators on the SGA projects also included Michael Artin ( étale cohomology ), Nick Katz ( monodromy theory , and Lefschetz pencils ). Jean Giraud worked out torsor theory extensions of nonabelian cohomology there as well. Many others such as David Mumford , Robin Hartshorne , Barry Mazur and C.P. Ramanujam were also involved. Alexander Grothendieck's work during what
9086-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of
9240-404: The reunification of East and West Germany to take place and resolved most post–World War II issues. No formal peace treaty between Japan and the Soviet Union was ever signed, although the state of war between the two countries was terminated by the Soviet–Japanese Joint Declaration of 1956 , which also restored full diplomatic relations between them. World War I had radically altered
9394-605: The strategic bombing of population centres and delivery of the only two nuclear weapons ever used in war. World War II was the deadliest conflict in history, resulting in 70 to 85 million deaths , more than half being civilians. Millions died in genocides , including the Holocaust of European Jews, as well as from massacres, starvation, and disease. Following the Allied powers' victory, Germany , Austria , Japan , and Korea were occupied, and war crimes tribunals were conducted against German and Japanese leaders . The causes of World War II included unresolved tensions in
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#17328583477589548-484: 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 . Grothendieck 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
9702-534: 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 . It also includes a study of Teichmüller theory . In 1983, stimulated by correspondence with Ronald Brown and Tim Porter at Bangor University , Grothendieck wrote
9856-409: The Allies were attempting to cut off . Denmark capitulated after six hours , and despite Allied support , Norway was conquered within two months. British discontent over the Norwegian campaign led to the resignation of Prime Minister Neville Chamberlain , who was replaced by Winston Churchill on 10 May 1940. On the same day, Germany launched an offensive against France . To circumvent
10010-400: The Axis war against the Soviet Union, in Romania's case partially to recapture territory ceded to the Soviet Union . In early June 1940, the Italian Regia Aeronautica attacked and besieged Malta , a British possession. From late summer to early autumn, Italy conquered British Somaliland and made an incursion into British-held Egypt . In October, Italy attacked Greece , but the attack
10164-421: The Blitz , but largely ended in May 1941 after failing to significantly disrupt the British war effort. Using newly captured French ports, the German Navy enjoyed success against an over-extended Royal Navy , using U-boats against British shipping in the Atlantic . The British Home Fleet scored a significant victory on 27 May 1941 by sinking the German battleship Bismarck . In November 1939,
10318-405: The European Axis declaring war on the US. Japan conquered much of coastal China and Southeast Asia , but its advances in the Pacific were halted in mid-1942 after its defeat in the naval Battle of Midway ; Germany and Italy were defeated in North Africa and at Stalingrad in the Soviet Union. Key setbacks in 1943—including German defeats on the Eastern Front, the Allied invasions of Sicily and
10472-410: The French possessions of Syria and Lebanon , assisted by the Free French . With the situation in Europe and Asia relatively stable, Germany, Japan, and the Soviet Union made preparations for war. With the Soviets wary of mounting tensions with Germany, and the Japanese planning to take advantage of the European War by seizing resource-rich European possessions in Southeast Asia , the two powers signed
10626-426: The Germans, and the last large operational unit of the Polish Army surrendered on 6 October . Despite the military defeat, Poland never surrendered; instead, it formed the Polish government-in-exile and a clandestine state apparatus remained in occupied Poland. A significant part of Polish military personnel evacuated to Romania and Latvia; many of them later fought against the Axis in other theatres of
10780-523: 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 . Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics. In 1970, Grothendieck, with two other mathematicians, Claude Chevalley and Pierre Samuel , created
10934-427: The IHÉS, those who knew him say that the causes of the rupture ran more deeply. 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. In that publication, Cartier notes that as the son of an antimilitary anarchist and one who grew up among the disenfranchised, Grothendieck always had
11088-518: The IHÉS. After leaving the IHÉS, Grothendieck became a temporary professor at Collège de France for two years. He then became a professor at the University of Montpellier, where he became increasingly estranged from the mathematical community. He formally retired in 1988, a few years after having accepted a research position at the CNRS . While not publishing mathematical research in conventional ways during
11242-590: The International Congress of Mathematicians (ICM) in Moscow, where he was to receive the Fields Medal. He retired from scientific life around 1970 after he had found out that IHÉS was partly funded by the military. He returned to academia a few years later as a professor at the University of Montpellier . While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from
11396-528: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became
11550-505: The Nationalists. Both Germany and the Soviet Union used this proxy war as an opportunity to test in combat their most advanced weapons and tactics. The Nationalists won the civil war in April 1939; Franco, now dictator, remained officially neutral during World War II but generally favoured the Axis . His greatest collaboration with Germany was the sending of volunteers to fight on the Eastern Front . In July 1937, Japan captured
11704-632: The Nations ". After the war, the young Grothendieck studied mathematics in France, initially at the University of Montpellier where at first he did not perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure . After three years of increasingly independent studies there, he went to continue his studies in Paris in 1948. Initially, Grothendieck attended Henri Cartan 's Seminar at École Normale Supérieure , but he lacked
11858-505: The Royal Navy putting three Italian battleships out of commission after a carrier attack at Taranto , and neutralising several more warships at the Battle of Cape Matapan . Italian defeats prompted Germany to deploy an expeditionary force to North Africa; at the end of March 1941, Rommel 's Afrika Korps launched an offensive which drove back Commonwealth forces. In less than a month, Axis forces advanced to western Egypt and besieged
12012-584: The Soviet Union occupied the entire territories of Estonia, Latvia and Lithuania, as well as the Romanian regions of Bessarabia, Northern Bukovina, and the Hertsa region . In August 1940, Hitler imposed the Second Vienna Award on Romania which led to the transfer of Northern Transylvania to Hungary. In September 1940, Bulgaria demanded Southern Dobruja from Romania with German and Italian support, leading to
12166-780: The Soviet Union , opening the Eastern Front and initially making large territorial gains. Japan aimed to dominate East Asia and the Asia-Pacific , and by 1937 was at war with the Republic of China . In December 1941, Japan attacked American and British territories in Southeast Asia and the Central Pacific , including Pearl Harbor in Hawaii , which resulted in the US and the UK declaring war against Japan, and
12320-621: The Soviet Union and Mongolia . The Japanese doctrine of Hokushin-ron , which emphasised Japan's expansion northward, was favoured by the Imperial Army during this time. This policy would prove difficult to maintain in light of the Japanese defeat at Khalkin Gol in 1939, the ongoing Second Sino-Japanese War and ally Nazi Germany pursuing neutrality with the Soviets. Japan and the Soviet Union eventually signed
12474-738: The Soviet Union should be eliminated and aimed for the conquest of Ukraine , the Baltic states and Byelorussia . However, other senior German officials like Ribbentrop saw an opportunity to create a Euro-Asian bloc against the British Empire by inviting the Soviet Union into the Tripartite Pact. In November 1940, negotiations took place to determine if the Soviet Union would join the pact. The Soviets showed some interest but asked for concessions from Finland, Bulgaria, Turkey, and Japan that Germany considered unacceptable. On 18 December 1940, Hitler issued
12628-402: The US dropped the first atomic bombs on Hiroshima and Nagasaki on 6 and 9 August. Faced with an imminent invasion of the Japanese archipelago , the possibility of further atomic bombings, and the Soviet declaration of war against Japan and its invasion of Manchuria , Japan announced its unconditional surrender on 15 August and signed a surrender document on 2 September 1945 , marking
12782-568: The United Kingdom and France guaranteed their support for Polish independence ; when Italy conquered Albania in April 1939, the same guarantee was extended to the Kingdoms of Romania and Greece . Shortly after the Franco - British pledge to Poland, Germany and Italy formalised their own alliance with the Pact of Steel . Hitler accused the United Kingdom and Poland of trying to "encircle" Germany and renounced
12936-538: The United Kingdom attacked on 3 July in an attempt to prevent its seizure by Germany. The air Battle of Britain began in early July with Luftwaffe attacks on shipping and harbours . The German campaign for air superiority started in August but its failure to defeat RAF Fighter Command forced the indefinite postponement of the proposed German invasion of Britain . The German strategic bombing offensive intensified with night attacks on London and other cities in
13090-455: The United Kingdom. The Germans turned south against the weakened French army, and Paris fell to them on 14 June. Eight days later France signed an armistice with Germany ; it was divided into German and Italian occupation zones , and an unoccupied rump state under the Vichy Regime , which, though officially neutral, was generally aligned with Germany. France kept its fleet, which
13244-640: The United States was assisting China and the Western Allies, and had amended the Neutrality Act to allow "cash and carry" purchases by the Allies. In 1940, following the German capture of Paris, the size of the United States Navy was significantly increased . In September the United States further agreed to a trade of American destroyers for British bases . Still, a large majority of the American public continued to oppose any direct military intervention in
13398-590: The Yellow River ; this manoeuvre bought time for the Chinese to prepare their defences at Wuhan , but the city was taken by October. Japanese military victories did not bring about the collapse of Chinese resistance that Japan had hoped to achieve; instead, the Chinese government relocated inland to Chongqing and continued the war. In the mid-to-late 1930s, Japanese forces in Manchukuo had sporadic border clashes with
13552-489: The ablest French and other mathematicians of the younger generation). Grothendieck practically ceased publication of papers through the conventional, learned journal route. However, he was able to play a dominant role in mathematics for approximately a decade, gathering a strong school. Officially during this time, he had as students Michel Demazure (who worked on SGA3, on group schemes ), Luc Illusie (cotangent complex), Michel Raynaud , Jean-Louis Verdier (co-founder of
13706-593: The anarchist militia. In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterward his father was interned in Le Vernet . He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners." The first camp was the Rieucros Camp , where his mother contracted the tuberculosis that would eventually cause her death in 1957. While there, Grothendieck managed to attend
13860-481: The application of L spaces in studying linear maps between topological vector spaces. In a few years, he had become a leading authority on this area of functional analysis—to the extent that Dieudonné compares his impact in this field to that of Banach . It is, however, in algebraic geometry and related fields where Grothendieck did his most important and influential work. From approximately 1955 he started to work on sheaf theory and homological algebra , producing
14014-491: The beginning of 1955, and there he set his old subject aside in order to work in algebraic topology and homological algebra , and increasingly in algebraic geometry. It was in Lawrence that Grothendieck developed his theory of abelian categories and the reformulation of sheaf cohomology based on them, leading to the very influential " Tôhoku paper ". In 1957 he was invited to visit Harvard University by Oscar Zariski , but
14168-471: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. World War II Asia-Pacific Mediterranean and Middle East Other campaigns Coups World War II or the Second World War (1 September 1939 – 2 September 1945)
14322-502: The best universal foundation for this field, because of its expressiveness as well as its technical depth. In that setting one can use birational geometry , techniques from number theory , Galois theory , commutative algebra , and close analogues of the methods of algebraic topology , all in an integrated way. Grothendieck is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation. Relatively little of his work after 1960
14476-456: The capital Nanking in December 1937. After the fall of Nanking, tens or hundreds of thousands of Chinese civilians and disarmed combatants were murdered by the Japanese . In March 1938, Nationalist Chinese forces won their first major victory at Taierzhuang , but then the city of Xuzhou was taken by the Japanese in May. In June 1938, Chinese forces stalled the Japanese advance by flooding
14630-403: The conflict well into 1941. In December 1940, Roosevelt accused Hitler of planning world conquest and ruled out any negotiations as useless, calling for the United States to become an " arsenal of democracy " and promoting Lend-Lease programmes of military and humanitarian aid to support the British war effort; Lend-Lease was later extended to the other Allies, including the Soviet Union after it
14784-515: The creation of a " New Roman Empire ". Adolf Hitler , after an unsuccessful attempt to overthrow the German government in 1923, eventually became the Chancellor of Germany in 1933 when Paul von Hindenburg and the Reichstag appointed him. Following Hindenburg's death in 1934, Hitler proclaimed himself Führer of Germany and abolished democracy, espousing a radical, racially motivated revision of
14938-739: The dissolution of the Austro-Hungarian, Ottoman, and Russian Empires. To prevent a future world war, the League of Nations was established in 1920 by the Paris Peace Conference . The organisation's primary goals were to prevent armed conflict through collective security, military, and naval disarmament , as well as settling international disputes through peaceful negotiations and arbitration. Despite strong pacifist sentiment after World War I , irredentist and revanchist nationalism had emerged in several European states. These sentiments were especially marked in Germany because of
15092-683: The end of May completed the German conquest of the Balkans. Partisan warfare subsequently broke out against the Axis occupation of Yugoslavia , which continued until the end of the war. In the Middle East in May, Commonwealth forces quashed an uprising in Iraq which had been supported by German aircraft from bases within Vichy-controlled Syria . Between June and July, British-led forces invaded and occupied
15246-523: The end of the century would lead to an unprecedented collapse of civilization. Grothendieck added however that his views were "in no way meant as a criticism of the Royal Academy's aims in the administration of its funds" and he added, "I regret the inconvenience that my refusal to accept the Crafoord prize may have caused you and the Royal Academy." La Clef des Songes , a 315-page manuscript written in 1987,
15400-463: The end of the war. World War II changed the political alignment and social structure of the world, and it set the foundation of international relations for the rest of the 20th century and into the 21st century. The United Nations was established to foster international cooperation and prevent conflicts, with the victorious great powers—China, France, the Soviet Union, the UK, and the US—becoming
15554-425: 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 . He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he
15708-566: The first step of what its government saw as the country's right to rule Asia , staged the Mukden incident as a pretext to invade Manchuria and establish the puppet state of Manchukuo . China appealed to the League of Nations to stop the Japanese invasion of Manchuria. Japan withdrew from the League of Nations after being condemned for its incursion into Manchuria. The two nations then fought several battles, in Shanghai , Rehe and Hebei , until
15862-470: The first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c. 582 – c. 507 BC ) established the Pythagorean school , whose doctrine it
16016-442: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of
16170-936: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,
16324-843: The former Chinese imperial capital of Peking after instigating the Marco Polo Bridge incident , which culminated in the Japanese campaign to invade all of China. The Soviets quickly signed a non-aggression pact with China to lend materiel support, effectively ending China's prior cooperation with Germany . From September to November, the Japanese attacked Taiyuan , engaged the Kuomintang Army around Xinkou , and fought Communist forces in Pingxingguan . Generalissimo Chiang Kai-shek deployed his best army to defend Shanghai , but after three months of fighting, Shanghai fell. The Japanese continued to push Chinese forces back, capturing
16478-414: The fourth volume (Part IV) was completed and, although unpublished, copies of it as a typed manuscript are circulated. Grothendieck helped with the translation and wrote a preface for it, in which he called Tsuji his "first true collaborator". Parts of Récoltes et Semailles have been translated into Spanish, as well as into a Russian translation that was published in Moscow. In 1988, Grothendieck declined
16632-629: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"
16786-561: 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 may be defined as certain derived functors in this context. Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray . Grothendieck took them to
16940-533: The initial Mathematische Arbeitstagung in Bonn , in 1957. It appeared in print in a paper written by Armand Borel with Serre. This result was his first work in algebraic geometry. Grothendieck went on to plan and execute a programme for rebuilding the foundations of algebraic geometry, which at the time were in a state of flux and under discussion in Claude Chevalley 's seminar. He outlined his programme in his talk at
17094-512: The invasion of the Ethiopian Empire (also known as Abyssinia ) by the armed forces of the Kingdom of Italy ( Regno d'Italia ), which was launched from Italian Somaliland and Eritrea . The war resulted in the military occupation of Ethiopia and its annexation into the newly created colony of Italian East Africa ( Africa Orientale Italiana , or AOI); in addition it exposed the weakness of
17248-466: 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, by means of a categorical Galois theory , an algebraic definition of fundamental groups of schemes giving birth to
17402-520: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at
17556-421: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to
17710-483: The local school, at Mendel. Once, he managed to escape from the camp, intending to assassinate Hitler . Later, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II . Grothendieck was permitted to live separated from his mother. In the village of Le Chambon-sur-Lignon , he was sheltered and hidden in local boarding houses or pensions , although he occasionally had to seek refuge in
17864-785: The mathematical community and devoted himself to political and religious pursuits (first Buddhism and later, a more Catholic Christian vision). In 1991, he moved to the French village of Lasserre in the Pyrenees , where he lived in seclusion, still working on mathematics and his philosophical and religious thoughts until his death in 2014. Grothendieck was born in Berlin to anarchist parents. His father, Alexander "Sascha" Schapiro (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
18018-497: The mathematical community, a community that initially accepted him in an open and welcoming manner, but which he progressively perceived to be governed by competition and status. He complains about what he saw as the "burial" of his work and betrayal by his former students and colleagues after he had left the community. Récoltes et Semailles was finally published in 2022 by Gallimard and, thanks to French science historian Alain Herreman,
18172-601: The mathematical establishment to come into contact with him". After his death, it was revealed that he lived alone in a house in Lasserre, Ariège , a small village at the foot of the Pyrenees . In January 2010, Grothendieck wrote the letter entitled "Déclaration d'intention de non-publication" to Luc Illusie , claiming that all materials published in his absence had been published without his permission. He asked that none of his work be reproduced in whole or in part and that copies of this work be removed from libraries. He characterized
18326-455: The necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil , he moved to the University of Nancy where two leading experts were working on Grothendieck's area of interest, topological vector spaces : Jean Dieudonné and Laurent Schwartz . The latter had recently won a Fields Medal. Dieudonné and Schwartz showed the new student their latest paper La dualité dans les espaces ( F ) et ( LF ) ; it ended with
18480-538: The notes of the Séminaire de géométrie algébrique ( SGA ) that he directed at the IHÉS. Grothendieck's political views were radical and pacifistic . He strongly opposed both United States intervention in Vietnam and Soviet military expansionism . To protest against the Vietnam War , he gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed. In 1966, he had declined to attend
18634-453: The now famous étale fundamental group and he then conjectured the existence of a further generalization of it, which is now known as the fundamental group scheme . As a framework for his coherent duality theory, he also introduced derived categories , which were further developed by Verdier. The results of his work on these and other topics were published in the EGA and in less polished form in
18788-556: The offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government—a refusal which, he was warned, threatened to land him in prison. The prospect of prison did not worry him, so long as he could have access to books. Comparing Grothendieck during his Nancy years to the École Normale Supérieure -trained students at that time ( Pierre Samuel , Roger Godement , René Thom , Jacques Dixmier , Jean Cerf , Yvonne Bruhat , Jean-Pierre Serre , and Bernard Malgrange ), Leila Schneps said: He
18942-497: The permanent members of its security council . The Soviet Union and the United States emerged as rival superpowers , setting the stage for the Cold War . In the wake of European devastation, the influence of its great powers waned, triggering the decolonisation of Africa and Asia . Most countries whose industries had been damaged moved towards economic recovery and expansion . World War II began in Europe on 1 September 1939 with
19096-571: The political European map with the defeat of the Central Powers —including Austria-Hungary , Germany , Bulgaria , and the Ottoman Empire —and the 1917 Bolshevik seizure of power in Russia , which led to the founding of the Soviet Union. Meanwhile, the victorious Allies of World War I , such as France, Belgium, Italy, Romania, and Greece, gained territory, and new nation-states were created out of
19250-583: The port of Tobruk . By late March 1941, Bulgaria and Yugoslavia signed the Tripartite Pact ; however, the Yugoslav government was overthrown two days later by pro-British nationalists. Germany and Italy responded with simultaneous invasions of both Yugoslavia and Greece , commencing on 6 April 1941; both nations were forced to surrender within the month. The airborne invasion of the Greek island of Crete at
19404-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in
19558-419: 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. He went on to introduce the étale cohomology theory of schemes, providing
19712-485: The proposal Esquisse d'un Programme ("Sketch of a Programme") for a position at the Centre National de la Recherche Scientifique (CNRS). It describes new ideas for studying the moduli space of complex curves. Although Grothendieck never published his work in this area, the proposal inspired other mathematicians to work in the area by becoming the source of dessin d'enfant theory and anabelian geometry . Later, it
19866-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate
20020-693: The remainder of Czechoslovakia and subsequently split it into the German Protectorate of Bohemia and Moravia and a pro-German client state , the Slovak Republic . Hitler also delivered an ultimatum to Lithuania on 20 March 1939, forcing the concession of the Klaipėda Region , formerly the German Memelland . Greatly alarmed and with Hitler making further demands on the Free City of Danzig ,
20174-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced
20328-410: The significant territorial, colonial, and financial losses imposed by the Treaty of Versailles . Under the treaty, Germany lost around 13 percent of its home territory and all its overseas possessions , while German annexation of other states was prohibited, reparations were imposed, and limits were placed on the size and capability of the country's armed forces . The German Empire was dissolved in
20482-447: The start or prelude to World War II. The exact date of the war's end also is not universally agreed upon. It was generally accepted at the time that the war ended with the armistice of 15 August 1945 ( V-J Day ), rather than with the formal surrender of Japan on 2 September 1945, which officially ended the war in Asia . A peace treaty between Japan and the Allies was signed in 1951. A 1990 treaty regarding Germany's future allowed
20636-415: The strong Maginot Line fortifications on the Franco-German border, Germany directed its attack at the neutral nations of Belgium , the Netherlands , and Luxembourg . The Germans carried out a flanking manoeuvre through the Ardennes region, which was mistakenly perceived by the Allies as an impenetrable natural barrier against armoured vehicles. By successfully implementing new Blitzkrieg tactics,
20790-415: The two wars became World War II in 1941. Other proposed starting dates for World War II include the Italian invasion of Abyssinia on 3 October 1935. The British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the forces of Mongolia and the Soviet Union from May to September 1939. Others view the Spanish Civil War as
20944-407: The war were not fulfilled in the peace settlement. From 1922 to 1925, the Fascist movement led by Benito Mussolini seized power in Italy with a nationalist, totalitarian , and class collaborationist agenda that abolished representative democracy, repressed socialist, left-wing, and liberal forces, and pursued an aggressive expansionist foreign policy aimed at making Italy a world power, promising
21098-475: The war. Germany annexed western Poland and occupied central Poland ; the Soviet Union annexed eastern Poland ; small shares of Polish territory were transferred to Lithuania and Slovakia . On 6 October, Hitler made a public peace overture to the United Kingdom and France but said that the future of Poland was to be determined exclusively by Germany and the Soviet Union. The proposal was rejected and Hitler ordered an immediate offensive against France, which
21252-404: The west halted the German advance for several days, but it was outflanked and encircled by the Wehrmacht . Remnants of the Polish army broke through to besieged Warsaw . On 17 September 1939, two days after signing a cease-fire with Japan , the Soviet Union invaded Poland under the supposed pretext that the Polish state had ceased to exist. On 27 September, the Warsaw garrison surrendered to
21406-415: The woods during Nazi raids, surviving at times without food or water for several days. His father was arrested under the Vichy anti-Jewish legislation , and sent to the Drancy internment camp , 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. In Le Chambon, Grothendieck attended the Collège Cévenol (now known as
21560-483: The world order , and soon began a massive rearmament campaign . France, seeking to secure its alliance with Italy, allowed Italy a free hand in Ethiopia , which Italy desired as a colonial possession. The situation was aggravated in early 1935 when the Territory of the Saar Basin was legally reunited with Germany, and Hitler repudiated the Treaty of Versailles, accelerated his rearmament programme, and introduced conscription. The United Kingdom, France and Italy formed
21714-888: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in
21868-554: Was invaded by Germany. The United States started strategic planning to prepare for a full-scale offensive against Germany. At the end of September 1940, the Tripartite Pact formally united Japan, Italy, and Germany as the Axis powers . The Tripartite Pact stipulated that any country—with the exception of the Soviet Union—that attacked any Axis Power would be forced to go to war against all three. The Axis expanded in November 1940 when Hungary , Slovakia , and Romania joined. Romania and Hungary later made major contributions to
22022-410: Was a global conflict between two coalitions: the Allies and the Axis powers . Nearly all the world's countries —including all the great powers —participated, with many investing all available economic, industrial, and scientific capabilities in pursuit of total war , blurring the distinction between military and civilian resources. Tanks and aircraft played major roles , with the latter enabling
22176-410: Was a three-volume collection of research papers to mark his sixtieth birthday in 1988. More than 20,000 pages of Grothendieck's mathematical and other writings are held at the University of Montpellier and remain unpublished. They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal. In 1991, Grothendieck moved to
22330-474: Was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received the Fields Medal in 1966 for advances in algebraic geometry , homological algebra , and K-theory . He later became professor at the University of Montpellier and, while still producing relevant mathematical work, he withdrew from
22484-432: Was claimed to have survived on the Holy Eucharist alone, Grothendieck almost starved himself to death in 1988. His growing preoccupation with spiritual matters was also evident in a letter entitled 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. The Grothendieck Festschrift , published in 1990,
22638-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support
22792-403: Was postponed until the spring of 1940 due to bad weather. After the outbreak of war in Poland, Stalin threatened Estonia , Latvia , and Lithuania with military invasion, forcing the three Baltic countries to sign pacts allowing the creation of Soviet military bases in these countries; in October 1939, significant Soviet military contingents were moved there. Finland refused to sign
22946-507: 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 . Although lauded as "the Einstein of mathematics", his work also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems. The bulk of Grothendieck's published work
23100-532: Was published in two-volumes and entitled 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 extensive autobiographical work, Récoltes et Semailles ('Harvests and Sowings', 1986), Grothendieck describes his approach to mathematics and his experiences in
23254-657: Was repulsed with heavy Italian casualties; the campaign ended within months with minor territorial changes. To assist Italy and prevent Britain from gaining a foothold, Germany prepared to invade the Balkans, which would threaten Romanian oil fields and strike against British dominance of the Mediterranean. In December 1940, British Empire forces began counter-offensives against Italian forces in Egypt and Italian East Africa . The offensives were successful; by early February 1941, Italy had lost control of eastern Libya, and large numbers of Italian troops had been taken prisoner. The Italian Navy also suffered significant defeats, with
23408-441: Was required to go through the bureaucracy of the League of Nations, which rendered it essentially toothless. The United States, concerned with events in Europe and Asia, passed the Neutrality Act in August of the same year. Hitler defied the Versailles and Locarno Treaties by remilitarising the Rhineland in March 1936, encountering little opposition due to the policy of appeasement . In October 1936, Germany and Italy formed
23562-442: Was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom is all the more incredible; quite unique in the history of mathematics. His first works on topological vector spaces in 1953 have been successfully applied to physics and computer science, culminating in
23716-426: Was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c. AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of
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