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Home , Alexander Grothendieck, Fields Medal, Grigori Perelman

Heng Li

Alexander Grothendieck’s life

Alexander Grothendieck was a German born French . Grothendieck was born on March 28, 1928 in . His parents were Johanna Grothendieck and Alexander Schapiro. When he was five years old, Hitler became the Chancellor of the German Reich, and called to boycott all Jewish businesses, and ordered civil servants who were not of the Aryan race to retire. Grothendieck’s father was a Russian Jew and at that time, he used the name Tanaroff to hide his Jewish Identity. Even with a Russian name it is still too dangerous for Jewish people to stay in Berlin. In May 1933, his father, Alexander Schapiro, left to because the rise of . In December of the same year, his mother left Grothendieck with a foster family Heydorns in , and joined Schapiro in Paris.

While Alexander Grothendieck was with his foster family, Grothendieck’s parents went to Spain and partici- pated in the . After the Spanish Civil War, Johanna(Hanka) Grothendieck and Alexander Schapiro returned to . In Hamburg, Alexander Grothendieck attended to elementary schools and stud- ied at the Gymnasium (a secondary school). The Heydorns family are a part of the resistance against Hitler; they consider that it was too dangerous for young Grothendieck to stay in Germany, so in 1939 the Heydorns sent him to France to join his parents.

However, because of the outbreak of World War II all German in France were required by law to be sent to special internment camps. Grothendieck and his parents were arrested and sent to the camp, but fortunately young Alexander Grothendieck was allowed to continue his education at a village school that was a few miles away from the camp. After the Nazis invaded France, Grothendieck and his parents were sent to different camps. Grothendieck’s father was sent to Camp du Vernet. Luckily, Grothendieck was allowed to stay with his mother in Camp de Rieucros, a camp for women. However, while Alexander Grothendieck was in the Rieucros internment camp, somehow, he managed to attend to the College Cevenol, which continued his education. Within a few months after Germany invaded France, France was then completely taken over by . Unfortunately, in 1942 his father was captured and sent to the Auschwitz concentration camp by and was murdered by Nazis in the summer of the same year.

After World War II, Grothendieck moved to . Grothendieck received some scholarships. With these scholarships, he went to the University of Montpellier to study . After he graduated from the University of Montpellier, he went to Paris and attended many top ’ lectures and seminars. At that time, he was more interested in topological vector spaces than Algebraic .

After he got this bachelor’s degree in the University of Montpellier, he was advised by Elie Cartan, a famous French mathematician, to finish his doctorate degree at the University of Nancy. During that time, he wrote his dissertation on . After he finished his doctorate degree at University of Nancy, he join a mathematicians’ collective called . In the mid 1950s, he went to the University of Sao Paulo and the . While he was at the University of Kansas, he published his Tohoku paper, which has revolutionized the subject of .

In 1959, Grothendieck joined the Institut des Hautes Etudes (IHES), which was where he found his interest in algebraic and . In the IHES, his contribution to the and algebraic geometry have revolutionized the field. During the time at IHES, he has published ”The Elements de geometrie algebrique” and ”Fondements de la Geometrie Algebrique,” which are his most important publi- cations. Also, in IHES his seminar about Algebraic Geometry was also published as a book by the institution, which also has a huge influence on Algebraic Geometry. The time when Alexander Grothendieck was in IHES, which was also called his golden age. Alexander Grothendieck in 1966 when he won the

In 1966, Grothendieck won the world’s highest honor in mathematics: the Fields Medal for his work done in on Weil , , and the idea of the K-theory has completely revolutionized the field of algebraic geometry and homological algebra . The ceremony was held in that year, but Grothendieck refused to attend to the ceremony as a protest because of the Soviet’s aggressive military activities in Europe.

Started in 1960s, Grothendieck started to participate in many anti-war movements. He did not only protest the , but he was also involved in many political activities against NATO. In November 1967, he went to Northern in the middle of the to protest.

Alexander Grothendieck is giving homology lecture in Vietnam

After he came back from Vietnam, he gave several lectures and speeches about his trip to Vietnam and the destruction that he had seen. However, his efforts made only his advisor, , start to care about the Vietnam War. Other than Schwartz, Grothendieck was not able to convince other people to be as involved in the war. In 1970, Grothendieck was participating in a violent protest and got arrested because he hit two police officers.

2 That same year after he was arrested, Grothendieck discovered that the IHES was partially funded by the military of France. He left the IHES because he failed to change the fact that IHES was getting funds from the military department. After this, he announced that he would not work on mathematics any further. He became a professor in the institutions, but he started to not to give lecture on math. Instead, he talked about the social and peace related topics. When many universities invited him to give a lecture about mathematics, he will not go unless they agree that he can give a lecture about his political philosophies first. Later, he participated in a political movement, and was put on trial. As a result, he was sentenced for 6 months to prison, and had to pay a heavy fine. Luckily the six months in prison penalty was suspended, however, he still has to pay the heavy fine.

Alexander Grothendieck in 1988

In 1988, Grothendieck was awarded with the due to his achievement on the Weil’s conjecture. However, he declined the prize and wrote an open letter to the media. In the letter, he questioned the ethic of community, and criticized and mathematicians who only focus on their research and ignoring the fact that their funds are from the military. Within the letter, he also apologized to the Royal Swedish Academy of Sciences for rejecting the prize. In the same year, he retired and totally isolated himself from the mathematical community.

During the 80s, he also started to have interest in religions. First, he was very interested in Buddhism. However, that did not last very long because in the 80s he started to look into Christianity. At the same time, he start to have some psychological issues. For a period of time, he identified himself as a Catholic nun and stated that he lived on the Eucharist alone for thirty years. Other than this, he had many other religious hallucinations.

In the summer of 1991, Grothendieck left his home without telling anyone. Ever since 1991, there were very few people who knew where he was, not even his family. He was trying to erase all of his files, records, and publications. He also tried to avoid any types of human contact. Until 2000, two young mathematicians found him in a small town when he was buying supplies. In the year 2005, another mathematician tried to find Grothendieck but failed. During the time when he was disappeared from public site, he wrote a huge amount of papers about philosophy, meditation, and the existence of evils. In the , he wrote hundreds of pages about the ”mutants.” According to his writings, these mutants are a of individuals who represented the best in the human race.

3 A very rare photo of Alexander Grothendieck in 2013.

Alexander Grothendieck passed away on November 13, 2014 at the age of 86. Until his death, Alexander Grothendieck remained stateless and did not have a French citizenship. Even though, he was a French math- ematician. Throughout his life, he traveled with a UN passport, not with a French passport because he had never applied for a French citizenship.

Alexander Grothendieck’s mathematical works

Perhaps the most important work that Alexander Grothendieck did is The Elements de geometrie algebrique or the Elements of Algebraic Geometry in English, also known as EGA. Grothendieck introduced the theory of schemes in the EGA. The theory of schemes is Grothendieck’s first huge significant mathematical work. In simplest terms, he proposed attaching to any a geometric object, called the Spectrum of the ring or an affine , and patching or gluing together these objects to form the scheme. The ring is to be thought of as the set of functions on its affine scheme. One of the purposes of scheme was to develop the formalism that’s needed for solving much deeper algebraic geometry problems. Once it was introduced, the theory has completely changed the field of Algebraic Geometry, because it became the universal framework or the language of Algebraic Geometry. Later on, the theory of scheme led to the proof of Fermat’s Last Theorem. Alexander Grothendieck worked with Jean Dieudonne together on the EGA. However, only a small part of the book was actually published.

4 Elements de geometrie algebrique(EGA)

Seminaire de Geometrie Algebrique du Bois Marie or SGA was a seminar that was taught by Alexander Grothendieck in IHES. Later on, the notes of the seminar were published by IHES as books. Perhaps, SGA was the another most important work of Alexander Grothendieck. In EGA, most parts of the book were not published, but SGA covered most unpublished parts. SGA has included countless ideas that were developed by Grothendieck. Later on, Deligne’s work on the Weil’s conjecture was directly related to the SGA.

Another publication that Grothendieck was known for was his Tohoku paper. In his Tohoku paper, he in- troduced the idea of treating a category as a class of object equipped with a class of . Which is different from the original view of Eilenberg and Mac Lan. In the paper, Grothendieck observed that modules over a ring and sheaves of abelian groups have the same behavior, which allows later mathematicians expand homological algebra in a cohesive way.

Alexander Grothendieck was also well known by his Grothendieck Riemann Roch Theorem(GRR). The GRR is a generalization of the classical Riemann Roch Theorem. This Theorem has a huge influence on the later development of the Atiyah Singer Index theorem.

Grothendieck is considered to be the greatest mathematician in the 20th Century. and , two famous American mathematicians once wrote:

Although mathematics became more and more abstract and general throughout the 20th century, it was Alexan- der Grothendieck who was the greatest master of this trend. His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed them- selves and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed.

Collaboration with other scholars

Throughout Alexander Grothendieck’s Career, most of his mathematical works were done by himself. So, there were only a few details of his collaborations with other mathematicians.

5 was one of this students, he finished his Doctoral Degree under Grothendieck’s supervision. They have collaborated in many topics, perhaps the most famous result of their works is the Weil conjecture. However, they did not collaborate directly on the Weil conjecture, but Deligne’s proof was largely extended based on Grothendieck’s SGA. Finally, in 1973, Deligne solved the last part of the Weil conjecture. In addition, both Deligne and Grothendieck were awarded the Crafoord Prize in 1988, on their achievement of the Weil’s Conjecture. According to the Crafoord Prize, Grothendieck and Deligne revolutionized algebraic geometry. Later developments in algebraic geometry (Faltings’ work about e.g. the big theorem of Fermat, Mori’s work about models of algebraic varieties and work by the prize winners and others) have demonstrated in a very convincing way the enormous potential and force of the ideas of the prizewinners. However, Grothendieck declined the prize due to his political philosophy.

Jean-Pierre Albert Achille Serre met with Alexander Grothendieck in the early 1950s. Alexander Grothendieck attended to Serre’s seminars, then they became friends with each other. Their collaborations led to many dis- coveries in homology, , and the important discoveries in algebraic geometry. During Grothendieck’s lifetime, they wrote many letters to each other, shared their ideas and discoveries. The two mathematicians’ works have totally revolutionized the field of algebraic geometry. Serre and Grothendieck also collaborated on Grothendieck Riemann Roch Theorem(GRR). There’re very few details about their collaborations, but GRR was originally conveyed in the letters from Grothendieck to Serre in the mid-1950s.

Jean Dieudonne is Grothendieck’s Doctoral advisor at the University of Nancy. They collaborated with each other on one of Grothendieck’s most important publications the Elements of Algebraic Geometry(EGA). EGA was published in the 1960s, by Grothendieck’s institution IHES. Grothendieck and Dieudonne planned thirteen chapters, but only the first four were actually published. However, the other chapters’ content can be found in the Seminaire de Geometrie Algebrique du Bois Marie(SGA).

In the mid-1950s, Grothendieck joined a very well known mathematicians association called Nicolas Bourbaki. In Bourbaki group, there are many other famous mathematicians such as Andre Weil, , etc. Grothendieck worked with these mathematicians. However, Grothendieck did not stay in the group very long because soon after he was offered a position in IHES.

Historical events that marked Alexander Grothendieck’s life.

Alexander Grothendieck’s childhood was under the rise of Nazism and the World War II. He was forced to separate with parents in 1933 when he was 5, because of the rise of Adolf Hitler and the anti-Semitism movement in Germany. His father was forced to move to Paris, because of his father’s Jewish identity. The same year, his mother also moved to Paris, which left him with his foster family. Throughout most of his childhood, he and his family were separated because of the political movements and wars.

Holocaust is one of the most terrible events that has ever happened in the history. More than 11 million people including 1.5 million children were killed in this horrifying tragedy. Grothendieck’s family were one of the victims of the Holocaust. After Grothendieck was sent to France by his foster family, the World War II breaks out. Germany’s invasion of Poland triggered France to declare war on Germany. After the breakout of the war, all Germans who stays in France were forced to relocate to special internment camps. Alexander Grothendieck and his parents were arrested and sent to go to internment camps. Unfortunately, in 1942 his father was sent to Auschwitz Concentration Camp, and was murdered in the summer the same year.

Alexander Grothendieck was involved in many political activities during the Cold War. His political philosophy is different from both the Western and the Soviet side. During the Cold War, he has protested to both sides in a various way. He was awarded the Fields Medal in 1966. The ceremony was held in Moscow, however, he refused to travel to Moscow because of the Soviets Union’s military activities in Eastern Europe. Other than

6 peaceful protests, he also involved in many violent strikes. He was arrested for hitting two policemen once in a violent protest. In another strike, he was put on trial and sentenced 6 months in prison.

Another historical event that has marked his life is the Vietnam war. Alexander Grothendieck went to Northern Vietnam in November 1967 during the Vietnam War. After he went back to France, he gave speeches and lectures about his trip in Vietnam. However, his effort did not raise many people’s awareness about the war.

Significant historical events around the world during Alexander Grothendieck’s life

There are four major historical events around the world during Alexander Grothendieck’s life: the Great Depression, World War II, Vietnam War, and the Cold War.

Grothendieck was born in 1928, just one year after his birth, American encountered her the most devastating and long lasting economy meltdown in the history. Every day, there were tens of thousands of Americans lose their job and shelters. Until Franklin D. Roosevelt was elected in 1933, more than 20 percent of America’s population was unemployed. Finally, in 1939 the economy got better, because of the broke out of the World War II kicked American industry and economy into high gear.

The Second World War is the most devastating conflict that has ever existed in the human history. There were more than 30 countries that directly fought in the war and caused more than 75 million deaths. There are two parts of World War II, one was fought in Asia and on Pacific islands, and the other one was fought in Europe.

The outbreak of the war in Europe started in September 1939 with Germany’s invasion of Poland. Within two years, Nazi Germany has taken over most of Europe. As the Nazis’ power expanded throughout the entire continent, Adolf Hitler’s Nazi Germany and its allies started the Holocaust in 1941; a genocide that has murdered over 11 million people, including 1.5 million children. On June 6, 1944, also known as D-Day, allies invaded France, and successfully liberated France. In January, the Soviets attacked Poland and pushed the front line all the way to Germany. On May 8, 1945, also known as V-E Day, Nazi Germany surrendered to the Allies, which ended the war in Europe.

The war in Asia started with the Japanese invasion of in 1937. In July 1937, Japan took over the former Chinese imperial capital, . After the war broke out, the Soviet Union quickly signed treaties with China to lend material supports, which ended China’s cooperation with Germany at that time. Even with material supports from the Soviets, China is too weak to defend Imperial Japan. After a few months, China has lost more than one-third of its territory. In November 1937, Japan took over the capital of China, Nanking; and more than three hundred thousand of Chinese civilians and disarmed soldiers were murdered by the Japanese. After the fall of Nanking, China relocated its capital to Chongqing and continued to fight with Japan. In December 1941, Japan attacked Pearl Harbor, Philippine, Thailand, Malaysia, and Hong Kong. These attacks forced the United States, Britain, Australia, and other Southeast Asian countries declared war on Japan. The United States fought countless battles with Japan on thousands of islands in the Pacific Ocean. After the victory in the battle of Guadalcanal, allies gained momentum and slowly defeated Japanese troops in southeast Asia. In August 1945, the United States dropped two atomic bombs on Hiroshima and Nagasaki. On August 15, 1945, Japan surrendered to the Allied Power, which officially ended the Second World War.

Cold War is the state of tension between the western world and the communists. The United States and the Soviets are allies during the World War II and fought together against the Axis. However, after the war, their relationship with each other has become very intense, because of their extremely different political systems.Between 1947 to 1991. Two years after the end of World War II, the Soviet Union was consolidating its control over Eastern Europe. On the other side, the United States is lending military and financial support

7 to the Western European countries and created NATO to against the Soviets. The Berlin Blockade occurs during 1948 to 1949 was the first major conflict during the Cold War. Afterward, with the Chinese Civil War, Korean War, and the Vietnam War the conflicts between two superpowers became more and more intense. The United States and the Soviets started the Arms Race by building powerful nuclear weapons and missiles, and the Race. With the construction of the Berlin Wall, Cuban Missile Crisis, and many other conflicts. Finally, in the 1970s, two superpowers became less aggressive to each other and accommodate to create a more stable international system. The Cold War was ended in 1991, with the dissolution of the Soviet Union the tension that has been around for half century finally got released.

The Vietnam War(1954-1975) was one of the major conflicts between the Communist and the Western world during the Cold War. The war was mostly fought between the communist government of Vietnam and the United States. During the 21 years of war, more than 3 million people were killed including more than fifty thousand Americans.

Significant mathematical progress during the Alexander Grothendieck’s lifetime

Alexander Grothendieck is one of the most important and dominant mathematicians in the 20th century. His work revolutionized the entire field of algebraic geometry. However, other than Grothendieck, there are also many other significant mathematical progresses that were made by other mathematicians in the 20th century.

Alan Turing is another important mathematician in the 20th century. He was most well known for his contributions to the breaking of the German enigma code, which has saved more than 14 millions of life and basically invented the first computer. Before the war, Turing was concentrated on simplifying Godel’s abstract proof. His method of doing the proof led to one of the most devastating ideas in the field of mathematics: there is no way of knowing the problem is solvable or unsolvable.

Paul Cohen an American mathematician best known for his works done one Cantor’s . In the 1960s, he proved that Cantor’s continuum hypothesis can be both true and false. This is a mind blowing idea. After his proof of continuum hypothesis, that every modern mathematical proof needs a statement that states whether this proof has used continuum hypothesis or not.

John Nash is a famous American mathematician well known for his Nash equilibrium, which has a fundamental contribution to the game theory. In the Nash equilibrium, John Nash proposed a new concept of situation analyzing and decision making, by taking into account other’s decisions. His Nash equilibrium was used in various fields such as military, market economics, computing, and even in sports games. Nash equilibrium’s contribution to the world’s economy helped him win the in 1994. John Nash was also known by the public because of the movie ”A Beautiful Mind.”

Fermat’s Last Theorem states that there are no three distinct positive integers a, b, c can satisfy the following equitation.

an + bn = cn

n is an integer that’s greater than 2.

Fermat’s Last Theorem was formulated in 1637. For more than 350 years no one was able to prove this theorem. Finally in 1995, , a British mathematician proved the Fermat’s Last Theorem. The

8 proof itself is more than 150 pages long and took over seven years of Wile’s research time. After he proved the theorem, he was knighted and received many other honors. In Weil’s proof of Fermat’s Last Theorem, Wile has used many theorem and techniques from the Alexander Grothendieck’s EGA. One can argue that without Alexander Grothendieck’s EGA, it unimaginable for the proof to exist.

Poincare Conjecture Another significant mathematical progress during Grothendieck’s lifetime is the proof of Poincare Conjecture. Poincare conjecture was first posed in the early 1900s and finally was proved by Russian mathematician . After Perelman has solved the conjecture, he was awarded one million dollars in recognition of his work, but he declined the prize.

Four Color Theorem was the first major mathematical theorem that was proved by using computers. The Four Color Theorem was first proposed by Francis Guthrie in 1852. The conjecture was first proved once in 1879, but in 1890 the proof was found incorrect. Finally, in 1976, the conjecture was proved by Kenneth Appel and Wolfgang Haken using computers. To prove this conjecture, it took them 1200 hours of computing time to test 1500 configurations.

To note, the Four Color Theorem is:

In any given separation of a plane into contiguous regions (called a ”map”) the regions can be colored using at most four colors so that no two adjacent regions have the same color.

Connections between history and the development of mathematics

The most important and influential event that has happened in Alexander Grothendieck’s lifetime was the second World War. The destruction of the mathematical and science communities due to World War II and the anti-semitism of the Nazi regime caused the focus of the world mathematics and science shift to the United States. Many world renowned scientists and mathematicians, such as Albert Einstein, Andre Weil, John Von Neumann, and Hermann Weyl moved from Europe to the United States due to the war.

During the World War II, Alan Turing was a British mathematician who built the Bombe to decipher German enigma code. He basically developed the world’s first computer. His invention has made a major contribution to the allies winning in the World War II. Because of his achievement in computer science, he became the founding father of the theoretical computer science and artificial intelligence.

After Alan Turing, the development of computing technology improved the ability of the mathematical com- munity to do massive computations. In 1952, a computer called SWAC identified the 13th Mersenne prime number: 2257 − 1 To note: Mersenne prime is the prime numbers that is one less than a power of 2:

P = 2n − 1

As of today, with computers mathematicians were able to find the 49th Mersenne prime number which has more than 22 million digits. The Four Color theorem was proved in 1976, which is the first major mathematical theorem that was proved by computers.

Remarks

Alexander Grothendieck was awarded the Fields Medal in 1966, for work done on Weil conjecture and his Tohoku Paper. According to the IMU, ” Alexander Gorthendieck’s Built on work of Weil and Zariski and

9 effected fundamental advances in algebraic geometry. He introduced the idea of K-theory (the Grothendieck groups and rings). Revolutionized homological algebra in his celebrated ’Tohoku paper’. ” However, because of the soviets military activities in Europe, he did not attend the ceremony that was held in Moscow.

He has also won the Crafoord Prize in 1988, but he declined the prize. Because he did not want the money, and because the award was in recognition of the work that he has done much earlier in the career. More important, he wants to draw the attention of the world because of the fact that the mathematics that was developed and used in pace are used for military-related purposes.

To recognize Alexander Grothendieck’s contributions to the institution and the mathematical community,IHES and the French National Center for Scientific Research(CNRS) created a research laboratory which names after Grothendieck called ”Alexander Grothendieck Laboratory”. Their main focus of the laboratory in mathematics is Algebraic geometry and , while the scope of the laboratory also covers theoretical such as String theory, Quantum field, and General relativity.

References

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15. https://webusers.imj-prg.fr/~leila.schneps/corr.pdf 16. https://mattbakerblog.wordpress.com/2014/11/15/excerpts-from-the-grothendieck-serre-correspondence/

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10 19. https://en.wikipedia.org/wiki/Alexander_Grothendieck

20. http://www.math.chalmers.se/~tamos/teaching/schemes.html 21. http://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html

22. http://math.stanford.edu/~brianrl/notes/schemes.pdf 23. http://www.landsburg.com/grothendieck/pragasz.pdf

24. https://en.wikipedia.org/wiki/Grothendieck?Riemann?Roch_theorem

25. https://en.wikipedia.org/wiki/Poincare_conjecture

26. https://ncatlab.org/nlab/show/Tohoku

27. http://www.ihes.fr/jsp/site/Portal.jsp?document_id=3744&portlet_id=1193

28. http://www.history.com/topics/great-depression

29. https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

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