Alexander Grothendieck facts for kids

Quick facts for kids Alexander Grothendieck
Alexander Grothendieck in Montreal, 1970
Born	(1928-03-28)28 March 1928 Berlin, Prussia, Germany
Died	13 November 2014(2014-11-13) (aged 86) Saint-Lizier, France
Nationality	French since 1971 Stateless (post 1945 until 1971)
Alma mater	University of Montpellier University of Nancy
Known for	Renewing algebraic geometry and synthesis between it and number theory and topology List of things named after Alexander Grothendieck
Awards	1966  Fields Medal 1977 Émile Picard Medal 1988  Crafoord Prize (declined)
Scientific career
Fields	Mathematics – functional analysis, algebraic geometry, homological algebra
Institutions	Institut des hautes études scientifiques (IHÉS) University of Montpellier University of São Paulo
Thesis	Produits tensoriels topologiques et espaces nucléaires (1953)
Doctoral advisors	Laurent Schwartz Jean Dieudonné
Doctoral students	Pierre Berthelot Pierre Deligne Michel Demazure Pierre Gabriel Jean Giraud Luc Illusie William Messing Michel Raynaud Hoàng Xuân Sính Jean-Louis Verdier

Alexander Grothendieck (born March 28, 1928 – died November 13, 2014) was a French mathematician. He is known as a very important figure in modern algebraic geometry. His work changed this field and brought in ideas from other areas of math. Many people consider him one of the greatest mathematicians of the 20th century.

Grothendieck started his public math career in 1949. In 1958, he became a research professor at the Institut des Hautes Études Scientifiques (IHÉS). He stayed there until 1970. He left because he disagreed with the institute getting money from the military. In 1966, he won the Fields Medal, a top award in math. He won it for his work in algebraic geometry and other math fields. Later, he taught at the University of Montpellier. He continued to do important math but also focused on political and religious ideas. In 1991, he moved to a small village in France. He lived there quietly, working on math and his thoughts until he died in 2014.

Life Story

Early Years and Family

Alexander Grothendieck was born in Berlin, Germany. His parents were anarchists, meaning they believed in a society without government. His father, Alexander Schapiro, had Jewish roots. His mother, Johanna Grothendieck, was a journalist from a Protestant family. Both parents had left their family backgrounds as teenagers.

When Alexander was born, his mother was married to another journalist. His birth name was first recorded as "Alexander Raddatz." His parents' marriage ended in 1929. Then, his father, Schapiro, said he was Alexander's dad. However, his parents never married. Alexander also had a half-sister named Maidi.

Alexander lived with his parents in Berlin until 1933. His father then moved to Paris to escape the Nazis. His mother followed soon after. Alexander was left with Wilhelm Heydorn, a teacher in Hamburg. His parents were involved in the Spanish Civil War.

Life During World War II

In May 1939, Grothendieck traveled by train to France. Soon after, his father was put in a camp called Le Vernet. From 1940 to 1942, Alexander and his mother were also held in camps. They were seen as "undesirable dangerous foreigners." At the Rieucros Camp, his mother got tuberculosis. This illness later caused her death.

While in the camp, Alexander went to a local school. Once, he tried to escape to try and assassinate Adolf Hitler. Later, his mother was moved to the Gurs internment camp. Alexander was allowed to live separately from her.

His father was arrested by the French Vichy government. He was sent to the Drancy internment camp. Then, he was handed over to the Germans. He was murdered at the Auschwitz concentration camp in 1942.

In the village of Le Chambon-sur-Lignon, Alexander found shelter. He stayed in local boarding houses. Sometimes, he had to hide in the woods during Nazi raids. He survived without food or water for days.

In Le Chambon, he attended the Collège Cévenol. This was a special school started by pacifists. Many refugee children went there. It was at this school that Alexander first became very interested in mathematics.

University Studies and Research

After the war, young Grothendieck studied math in France. He started at the University of Montpellier. At first, he did not do well and failed some classes. But by working on his own, he rediscovered an important math concept called the Lebesgue measure. After three years, he moved to Paris in 1948.

In Paris, he went to a seminar by Henri Cartan. But he did not have enough background for it. So, he moved to the University of Nancy. There, he studied with two famous mathematicians, Jean Dieudonné and Laurent Schwartz. Laurent Schwartz had just won a Fields Medal. Schwartz showed Grothendieck his latest paper. It had 14 unsolved problems. Grothendieck solved all of them in just a few months.

He wrote his doctoral paper (dissertation) in Nancy from 1950 to 1953. It was about functional analysis, a branch of math. By then, he was a top expert in this area. In 1953, he moved to Brazil to teach at the University of São Paulo. He used a Nansen passport because he refused to become a French citizen. He did not want to join the military. He stayed in Brazil until late 1954. His work there was still on functional analysis.

In 1955, Grothendieck moved to Lawrence, Kansas, in the USA. There, he started working on new math areas. These included algebraic topology and homological algebra. He also worked more and more on algebraic geometry. In Lawrence, he developed ideas that led to his famous "Tôhoku paper."

In 1957, he was invited to Harvard University. But he refused to sign a pledge. The pledge said he would not try to overthrow the U.S. government. So, the offer was canceled. He said he did not mind prison as long as he had books. His early math work has been used in physics and computer science.

Years at IHÉS

In 1958, Grothendieck joined the Institut des Hautes Études Scientifiques (IHÉS). This was a new research institute created for him and Jean Dieudonné. Grothendieck became very active there. He held many seminars, which were like intense study groups. He gathered many talented young mathematicians to work with him.

During this time, he mostly stopped publishing papers in regular journals. But he was a very important figure in math for about ten years. He had many students and collaborators. These included Pierre Deligne, Michael Artin, and Jean-Louis Verdier.

The "Golden Age" of Math

The time Grothendieck spent at IHÉS is called his "Golden Age." During this period, he connected many different areas of math. These included algebraic geometry, number theory, topology, and category theory.

One of his first big discoveries was the Grothendieck–Hirzebruch–Riemann–Roch theorem. This was a new version of an older math theorem. He also created K-theory, which studies how math objects relate to rings. Then, he developed the theory of schemes. This gave algebraic geometry a new, more flexible foundation.

He also created étale cohomology theory. This was a key tool for solving the Weil conjectures. These were big unsolved problems in math. He also started topos theory, which is a new way to think about topology. His work also led to derived categories.

The results of his work were published in huge collections. These were called Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA).

Political Beliefs and Activism

Grothendieck had strong political views. He was a pacifist and against war. He opposed the Vietnam War by the United States. He also opposed military actions by the Soviet Union. To protest the Vietnam War, he gave math lectures in the forests around Hanoi. This was while the city was being bombed.

In 1966, he refused to go to a big math conference in Moscow. He was supposed to receive the Fields Medal there. Around 1970, he left science. He found out that IHÉS received some funding from the military. This went against his beliefs. He returned to teaching a few years later at the University of Montpellier.

Leaving IHÉS was not just about military funding. Those who knew him said he deeply cared for the poor. He felt that IHÉS was like a "gilded cage." He also became interested in other scientific areas, like biology.

In 1970, Grothendieck started a political group called Survivre et vivre. It focused on anti-military and environmental issues. The group also criticized how science and technology were used. Grothendieck spent three years working for this group.

After leaving IHÉS, his regular math career mostly ended. He taught for two years at Collège de France. Then he became a professor at the University of Montpellier. He retired in 1988.

Writings in the 1980s

Even though he wasn't publishing in regular journals, Grothendieck wrote many important papers in the 1980s. These were shared with a small group of people.

One was La Longue Marche à travers la théorie de Galois. This was a 1600-page handwritten paper. It contained many new ideas.

In 1983, he wrote Pursuing Stacks, a 600-page paper. It was written like a diary. In it, he explained his ideas about how different math areas were connected. This paper led to another huge work, Les Dérivateurs, written in 1991.

In 1984, Grothendieck wrote Esquisse d'un Programme ("Sketch of a Programme"). This paper described new ways to study complex curves. It inspired other mathematicians to work on new theories.

He also allowed some of his older drafts to be published. These were from his EGA work.

In 1986, he wrote a 1,000-page autobiography called Récoltes et semailles. In it, he described his approach to math. He also wrote about his experiences in the math community. He felt that the community became too focused on competition. He also felt his work was ignored after he left. This book is now available online.

In 1988, Grothendieck turned down the Crafoord Prize. This is a very important award. He wrote that he and other famous mathematicians did not need more money. He also criticized what he saw as a decline in ethics in science. He believed that scientific theft was common. He also thought that civilization would collapse by the end of the century. But he added that he respected the Royal Academy's goals.

La Clef des Songes (1987) was a 315-page paper. In it, Grothendieck wrote about how dreams led him to believe in God. He also described 18 "mutants," people he admired as visionaries. The only mathematician on his list was Bernhard Riemann. He became very focused on spiritual matters. In 1990, he wrote a letter to friends. In it, he said he had met God and that a "New Age" would begin in 1996.

In 1990, a three-volume collection of research papers was published. It was called the Grothendieck Festschrift. It celebrated his sixtieth birthday.

More than 20,000 pages of Grothendieck's writings are at the University of Montpellier. They are available online for free.

Later Life and Death

In 1991, Grothendieck moved to a new home. He did not tell his old math contacts where he was. Very few people visited him after that. Local villagers helped him with food. He had tried to live only on dandelion soup.

In January 2010, Grothendieck wrote a letter. He said that any of his work published without his permission should be removed from libraries. He called a website about his work "an abomination." However, this request might have been changed later in 2010.

Alexander Grothendieck died on November 13, 2014. He was 86 years old. He lived alone in a house in Lasserre, Ariège, a small village in France.

Citizenship Status

Grothendieck was born in Germany. In 1938, he moved to France as a refugee. Records of his nationality were lost after World War II. He did not apply for French citizenship. So, he was a stateless person for most of his working life. He traveled using a Nansen passport. He did not want French nationality because it meant serving in the French military. He especially did not want to fight in the Algerian War. He finally applied for French citizenship in the early 1980s. By then, he was too old for military service.

His Family

Grothendieck was very close to his mother. He dedicated his doctoral paper to her. She died in 1957 from tuberculosis. She got the illness in camps for people who had lost their homes.

He had five children. He had one son with his landlady in Nancy. He had three children, Johanna, Alexander, and Mathieu, with his wife Mireille Dufour. He had one child with Justine Skalba.

Math Contributions

Early Math Work

Grothendieck's first math work was in functional analysis. From 1949 to 1953, he worked on his doctoral paper in this area. His teachers were Jean Dieudonné and Laurent Schwartz. He made important contributions to topological tensor products and nuclear spaces. In just a few years, he became a top expert in functional analysis.

Shift to Algebraic Geometry

However, Grothendieck's most important work was in algebraic geometry. Around 1955, he started working on sheaf theory and homological algebra. His famous "Tôhoku paper" (1957) introduced abelian categories. This helped define sheaf cohomology.

He used these new methods to change how algebraic geometry was done. He focused on the "relative point of view." This meant studying how different math objects relate to each other. This led to many classical theorems being generalized.

In 1956, he applied his ideas to the Riemann–Roch theorem. This led to the Grothendieck–Riemann–Roch theorem. This was his first big work in algebraic geometry. He then planned a huge project to rebuild the foundations of algebraic geometry. He presented his plan at a math conference in 1958.

His new foundation for algebraic geometry was very abstract. He introduced the theory of schemes. These are like special topological spaces with rings attached to them. Schemes became the basic objects studied in modern algebraic geometry. They allowed ideas from other math fields to be used in geometry.

Grothendieck was known for his very abstract way of thinking. He was also very careful about how he wrote his math. Most of his work after 1960 was not published in regular journals. Instead, it was shared in notes from his seminars. His influence came mostly from his personal teaching. His ideas also spread to many other areas of math.

EGA, SGA, FGA

Most of Grothendieck's published work is in two huge, but unfinished, collections. These are Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA). Another collection, Fondements de la Géometrie Algébrique (FGA), also has important material.

Grothendieck invented new cohomology theories. These theories use algebraic methods to study topological objects. They helped explain a connection between the shape of a math object and its number properties. For example, the number of solutions to an equation can show the topological nature of its solutions.

His work led to the proofs of the Weil conjectures. These were a set of big math problems. His student Pierre Deligne finally proved the last one in the 1970s. Grothendieck's approach was called a "visionary program." His cohomology theory became a key tool for number theorists.

Grothendieck also had a theory called motives. This was meant to be a general theory without choosing a specific prime number. It did not directly solve the Weil conjectures. But it led to new developments in other math areas.

Key Math Ideas

In his book Récoltes et Semailles, Grothendieck listed twelve of his ideas that he thought were "great." Here are some of them:

• Topological tensor products and nuclear spaces.
• "Continuous" and "discrete" duality.
• The Grothendieck–Riemann–Roch theorem and K-theory.
• Schemes.
• Topoi.
• Étale cohomology and l-adic cohomology.
• Motives.
• Crystals and crystalline cohomology.
• "Topological algebra" and derivators.
• Tame topology.
• Anabelian algebraic geometry.
• "Schematic" view for regular polyhedra.

He wrote that topoi was the biggest idea. It combined algebraic geometry, topology, and arithmetic. He also said that schemes were the most developed idea. They were the framework for many of his other ideas. He believed that motives and anabelian geometry were the deepest ideas.

Impact on Math

Grothendieck is seen by many as the greatest mathematician of the 20th century. By the 1970s, his work was very influential. It impacted not only algebraic geometry but also logic and category theory.

Geometry

Grothendieck changed algebraic geometry by making its foundations clearer. He also developed new math tools to solve important problems. Algebraic geometry traditionally studied geometric shapes using algebraic equations.

Grothendieck created a new foundation for this field. He made "spectra" (intrinsic spaces) and their related rings the main objects of study. He developed the theory of schemes. These are like topological spaces where a ring is linked to each part of the space. Schemes are now the basic objects studied in modern algebraic geometry.

His new version of the Riemann–Roch theorem connected the shapes of complex curves to their algebraic structure. This theorem is now named after him. The tools he created to prove this theorem started the study of algebraic and topological K-theory. These fields explore the shapes of objects by linking them to rings.

Cohomology Theories

Grothendieck created new cohomology theories. These theories use algebraic methods to study topological objects. This work influenced algebraic number theory, algebraic topology, and representation theory. As part of this, he created topos theory. This is a more general idea of topology. It has influenced set theory and mathematical logic.

The Weil conjectures were a set of math problems from the 1940s. They described properties of numbers related to algebraic curves. Grothendieck's discovery of étale cohomology helped solve these conjectures. His student Pierre Deligne completed the proof in the 1970s. Grothendieck's big-picture approach was called a "visionary program." His cohomology theory became a key tool for number theorists.

Grothendieck's idea of motives was meant to be a general theory. It did not directly lead to the Weil conjectures. But it has been important for modern developments in other math areas.

Category Theory

Grothendieck emphasized the idea of universal properties in math. This made category theory a main way to organize math in general. Category theory creates a common language. It helps describe similar structures and techniques found in many different math systems. His idea of abelian categories is now a basic object of study in homological algebra. Many people believe that Grothendieck's influence led to category theory becoming its own math field.

See Also

In Spanish: Alexander Grothendieck para niños

Images for kids

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  Alexander Grothendieck in Montreal, 1970