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Home , Alexander Grothendieck, Claude Chevalley, Descent (mathematics)

Kyle Aguilar

Alexander Grothendieck’s Life

Alexander Grothendieck was born on March 28, 1928. His parents names were Alexander Schapiro, a Russian Jew who fought in numerous revolutions against Russian Tsars, and Johanna Grothendieck, a German who met Schapiro at .

The family lived during the Hitler Era and there was a law passed stating if civil servants were not of Aryan , they were forced to retire. Because of this, in 1933, the father returned to and Grothendieck was arranged to live with a pastor named Wilhelm Heydorn and his wife in from 1934-1939. During this time, Grothendieck went to elementary school and would often study by himself in the gymnasium. Meanwhile, his parents traveled back to Spain to support the Republicans during the from 1936-1939. When the parents returned, a law passed stating that anyone of German descent were to be sent to internment camps. Generally speaking, because of Grothendieck’s race and the time era, he and his parents were sent to internment camps in . The mom and dad were separated, the mom staying with Grothendieck. Grothendieck was able to continue his education at a place that was five kilometers near the camp and also received private tutoring.

During Grothendieck’s studies, he was able to get into numerous colleges and universities, including University of , University of Nancy, University of Sao Paulo, University of Kansas, and Institut des Hautes Etudes Scientifiques, to study . In high school, he was very disappointed because of the absence of certain math (specifically geometric) definitions. He did not like that there was an absence of length of a curve, area of a surface, and volume of a solid. He thought going to college would change that. However, he did not find the education very helpful.

Grothendieck was exceling in mathematics with such ease, that all his professors would advise him to travel to various colleges and attend seminars. During his travels and studies, he was able to establish a number of theories that would soon help out future as well as become one of the members of the Bourbaki .

In addition to his works, he married a woman named Mireille and had three kids named Johanna, Mathieu, and Alexandre. He also received the in 1966. However, the presentation was held in Moscow and being a strong pacifist, he refused to travel to Moscow and let the director of one of the schools he went to receive the medal on his behalf. In rebellion, he went to North Vietnam in 1967 and lectured at a secret location in Hanoi.

Following that a few years later, he left Institut des Hautes Etudes Scientifiques, the school he was studying at during that time, because he discovered the school was receiving some funds from military sources. This led to him abandoning mathematics and changed to political protest - specifically on nuclear proliferation. Unlike his lectures in mathematics, the political protests were very ineffective.

He retired at sixty years old and his research was published in his honor. He was even awarded the in 1988 but declined it completely because he did not want the money and the award was for his work earlier in his career. During his retirement, he wrote thousands and thousands of pages (some math, some non-math) of his theorems and research. Then suddenly in 1991, he left home without informing anyone to an unknown location. In fact, in 2010, he tried to erase all traces of his life. He wrote a letter to one of his students to get rid of all his files and publications, refusing any of them to be republished. At age 86, November 13, 2014, Grothendieck passed away. Today he is considered one of the greatest mathematicians of the 20th century. Alexander Grothendieck’s Mathematical Works

Grothendieck developed a lot of theories and proofs over the course of his life. He focused his works on proving theories and solving difficult problems that other mathematicians he worked with couldn’t solve. Along the way, he was able to develop some of his own understandable concepts that helped future mathematicians. In fact, before Grothendieck made all these discoveries, his professor at the University of Montpellier and future collaborator, Elie Cartan, believed Lebesgue solved all outstanding mathematical problems. Note that Lebesgue was a who founded the theory of measure and generalized the notion of the Riemann integral. However, Grothendieck was able to formulate many theories.

First, while Grothendieck was studying at the University of Nancy, Schwartz and Dieudonne were trying to develop a general theory for locally convex spaces by studying Frchet spaces and their limits. However, they ran into some problems. When they consulted Grothendieck, he was able to solve all the problems in less than a year. Below is information about Frechet spaces.

Frechet spaces:

A Frechet (vector space) is “a complete and metrizable space, sometimes with the restriction that it can be locally convex”. As seen in this example, the space in between the two endpoints of the blue line is considered Frechet space.

This means:

T = (S, τ) is a if and only if

∀x, y ∈ S such that x 6= y. So that means:

∃U ∈ τ : x ∈ U, y∈ / U and ∃V ∈ τ : y ∈ V, x∈ / V

Generally speaking, this is saying that there are open sets of open space. There are no two sets of space that are the same.

2 Another definition:

Note that vector spaces such as Co(X) with non-compact X occur often.

V is a vector space with metric d(, )

Let d is translation invariant (meaning that the vector has translation symmetry), such that:

dx(x + z, y + z) = d(x, y) for x, y, z ∈ V The basis at v ∈ V consists of open balls centered at v

{w ∈ V : d(v, w) < r}

The translation invariance implies: v + Br = {v + b : b ∈ Br}

Br is an open ball with radius centered at 0. This will determine the of the whole vector space.

Topology on V is locally convex if there’s a local basis at every point (or by definition, 0) that has convex sets. A convex set contains elements from the vector space which all points between the two parts on the vector are part of the set. If the metric is a complete set (meaning it contains all points), then V is a Frechet space.

Grothendieck published his proof of Frechet spaces in his doctorial thesis call Produits tensoriels topologiques et espaces nucleaires. It’s currently published under the Bourbaki group. The paper he wrote was to demonstrate his understanding of another area of mathematics rather than just creating and finding new theories. According to sources, Grothendieck’s second thesis was on the theory (showing his interest in algebraic ), where he was supposed to show his greatest work. However, even the best of the best make mistakes and Cartan pointed out to him one day that there were some incorrect statements Grothendieck pointed out in his thesis.

The “Golden Age” of Grothendieck was his prime where he was able to make a lot more discoveries and proofs of different theories than he did in the past. These include the “theory of schemes” in the 1960s, work on the theory of topoi, algebraically proving Riemann-Roch’s theorem, an algebraic definition of the of a curve, etc. A few of these definitions of what he discovered are listed below:

Riemann-Roch Theorem

Definition: This theorem computes the dimension of the space of meromorphic functions with zeros and poles.

If D is a divisor on compact Riemann surface of genus g then

l(D) − i(D) = deg (D) + 1 − g

If D = 0 and substitute, then: l(0) − i(0) = deg (0) + 1 − g l(0) = 1 and deg(0) is 0 because it’s the sum of coefficients of the zero divisor. i(0) = l(K) L(K) is isomorphic to space. So: 1 − i(0) = 0 + 1 − g

3 and i(0) is equal to dimension of the space. And if we substitute D = K into the formula: l(K) − i(K) = deg (K) + 1 − g

And since l(K) = i(0) = g and i(K) = l(0) = 1 then g − 1 = deg (K) + 1 − g

So the formula is: deg (K) = 2g − 2

Grothendieck is also well known for his “Tohoko paper” in which he talks about numerous mathematical concepts including abelian categories, sheaves of modules, resolutions, derived functors, and the Grothendieck spectral sequence. In fact he was even rewarded the Fields medal for the advances in he showed in that paper.

Quoting from “The Grothendieck Festschrift” it stated all the work that Grothendieck did during his golden age saying,

“The mere enumeration of Grothendieck’s best known contributions is overwhelming: topological tensor prod- ucts and nuclear spaces, sheaf as derived functors, schemes, K-theory and Grothendieck-Riemann- Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck (sites) and topoi, derived categories, formalisms of local and global (the ’’), tale cohomology and the cohomological interpretation of L-functions, , ’standard conjectures’, motives and the ’yoga of weights’, tensor categories and motivic Galois groups. It is difficult to imagine that they all sprang from a single mind.”

Collaboration With Other Scholars

Grothendieck worked with numerous amounts of other scholars and mathematicians to help get where he was at. Some them were his professors, who after teaching him, advised him to go to other areas to the world to find out more about mathematics, and other scholars he taught in his own lectures and seminars.

First there was Elie Cartan, Grothendieck’s professor. He is a professor at the University of Montpellier who worked on continuous groups, , differential equations and geometry. Cartan advised Grothendieck to travel with him to Paris to attend his seminars and work with him. Of course, Grothendieck accepted the offer and followed. At the seminar, Grothendieck worked with many leading mathematicians during the twentieth century. These include , Jean Delsarte, Jean Dieudonne, , Laurent Schawartz, and Andre Weil.

Chevalley had a huge influence on many parts of math. He was able to come up with multiple theories as well including the and .

Delsarte was very big on the study of mathematics and applying that to scientific concepts. His doctoral theses were published by the Academy of Science he taught mathematical courses, and even some astronomy courses, at the University of Nancy.

Dieudonne was a professor who worked at various universities teaching mathematics. These include Sao Paulo, the University of Michigan, Northwestern University, Institut des Hautes Etudes Scientifiques, and a chair at the faculty of Science at Nice until 1970. Despite all the places Dieudonne taught at, his passion was more towards research than teaching other students.

4 Godement had a dream to become a school teacher. For the first nineteen years of his life, he stayed at home to help out his dad. Once he got into mathematics, he said that his life was saved by math. He’s a mathematician who still lives to this day and published multiple mathematical papers while he was a research student.

Laurent Schawartz had polio when he was eleven years old. Despite the fact that he recovered in just a few months, it made Schawartz weak his entire life. In his final years of studying, he had the choice of studying philosophy and humanities or mathematics and philosophy. He chose mathematics of course. During this time, he fell in love with the geometry aspect of math and fell out of philosophy. His fiance’s father, Paul Lvy, became a big influence on Schawartz. Lvy interests included probability theory and , two topics that Schawartz became interested in because of him.

Andre Weil was a French mathematician who focused his work on and . During his studies, he was fascinated by solving Diophantine equations and did his own mathematical research. In fact, Grothendieck’s work helped Weil make some interesting discoveries.

Grothendieck cooperated with all these mathematicians during the seminar and some even after that. Weil and Cartan advised Grothendieck to travel to Nancy where he collaborated with Dieudonne, Delsarte, Godement and Schwartz. Generally speaking, he helped Dieudonne in his research.

Grothendieck was also part of the well-known Bourbaki group in 1956. This group consisted of mathematicians he spent a lot of time working with: Cartan, Weil and Dieudonne. However, he did not stay long because he was offered a research position at the Institut des Hautes Etudes Scientifiques. So he went there to do research instead.

Historical Events that Marked Alexander Grothendieck’s Life

To start, Grothendieck was born on March 28, 1928. There was a time period where he stayed with a pastor because his parents went off to support the Republicans in the Spanish Civil War (from 1936-1939). In 1933, his mother arranged for him to be fostered by a pastor named Wilhelm Heydorn and his wife. However in 1939, the Heydorns were in serious danger because they became a resistance against Hitler. So Grothendieck traveled to Paris where he met up with his parents. In the summer, he would join his mother in the Nimes.

In 1938, the law passed where all “undesirables” (those of German descent) had to be sent to a special internment camp. Grothendieck and his mom went to one camp, and his dad was sent to another. Then in 1942, the camp that Grothendieck was in closed, so he and his mom were sent to Gurs concentration camp near Pau. Meanwhile, the father was sent over to the French Vichy government and the sent him to the Auschwitz extermination camp where he perished. Meanwhile, in 1945, Grothendieck was still receiving an education earned his baccalaurat at the Collge Cvnol. During this year, he and his mother moved to a village of Maisargues near the university he studied. Because of his outstanding performance at the university, he attended the cole Normale Suprieure in Paris from 1948-1949. This is where he attended one of his professor’s seminars and met many top notch mathematicians in the twentieth century.

Following in 1949, Grothendieck moved to the University of Nancy and lived with his mother who was fight- ing tuberculosis she obtained from the internment camps. During his time at Nancy, he would study with Dieudonne on functional analysis and attend weekly seminars on Saturday where they studied a variety of topics. In fact Dieudonne stated that Grothendieck helped solve the “general theory of duality for locally convex spaces in less than a year.

From 1953-1955, Grothendieck traveled to other universities to conduct more research. During this time, his interests changed more towards topology and geometry. His research was supported by Centre National de la Recherche Scientifique. This was the time when he became one of the famous Bourbaki groups of

5 mathematicians. He was teamed up with Weil, Cartan, and Dieudonne. But in 1959, he was offered another research opportunity at the Institut des Hautes Etudes Scientifiques (IHES) which he accepted.

This era, when he accepted Institut des Hautes Etudes Scientifiques’s offer to 1970, was considered the “Golden Age” of Grothendieck. He established a whole new school of mathematics and made interesting discoveries that helped other mathematicians in the future. During the 1950s, he married to a girl named Mirielle and had three kids: Johanna, Mathieu, and Alexandre.

In 1966, Grothendieck was awarded the Fields Medal for his mathematical discoveries, proofs and his well- known “Tohoko Paper” which consisted abelian categories, sheaves of modules, resolutions, derived functors, and the Grothendieck spectral sequence. There were difficulities giving the medal to him because the ceremony was all the way in Moscow. The problem is, Grothendieck was always a strong pacifist and campaigned against military build-up. So, he refused to travel to Moscow and the director of IHES received it on behalf of Grothendieck. In rebellion, he went to Vietnam in November and December of 1967 to teach math. This was during the time the Americans bombed the area. He taught in a secret area at Hanoi University for about a week and a half.

More emphasis on Grothendieck’s pacifism showed in 1970. During this time, he discovered that IHES was receiving some of their funds from military sources. Because of this, he resigned from IHES on May 25, 1970. Because of this he left mathematics and focused more on political protest. Unlike his math, his protests were very ineffective.

After teaching at multiple schools for some years, Grothendieck retired at age 60 in 1988. In honor of his retirement, his publications were produced. He was awarded the Crafoord Prize at this time too but he declined it. Reasons are unknown.

Between 1980-1990, Grothendieck wrote thousands upon thousands of pages containing all of his studies both math and non-math related. Then 1991 marked his disappearance. He told no one where he was off to, or when he left but he spent a lot of time writing a lot about .

November 13, 2014 marks the end of Grothendieck’s historical life.

Significant Historical Events Around the World During Alexander Grothendieck’s Life

A year after his birth (1929) to 1939, while the young Grothendieck was moving around from place to place with his family, the United States was going through a rough time as well. While he was moving around trying to stay safe, America was going through the infamous Great Depression. The stock market crashed in 1929 and people started to panic. Many people became unemployed, consumer spending dropped significantly, and industrial output decreased. Although this doesn’t directly affect Grothendieck, this shows that a lot of known parts of the world were struggling in different ways.

During Grothendieck’s childhood, he was a witness for World War II. As many people know, World War II was one of the most devastating wars in history with that had more than thirty countries involved and over fifty million deaths (both military and civilian based). Even before that, Grothendieck and his family had to go through Hitler’s reign in power because his mother was German and his father was a Jew.

In April 1933, Jewish shops and businesses were boycotted; this day was known to some as “boycott day”. In addition, the Nazis passed a law stating that commanded the retirement of civilians who were not of Aryan descent. This caused the family to run off to different places. The father went to Paris around May 1933 while the mother and Grothendieck went to Hamburg. Over in Hamburg, five year old Grothendieck was to be fostered by a pastor name Wilhelm Heydorn and wife. When the Heydorns were in danger in 1939, they

6 put him on a train to join his family in Paris.

During World War II, Grothendieck and his family was considered very dangerous in France and were required to stay in internment camps according to the new law that passed. This was because of suspicion that France had towards the Germans. They, including Britain, declared war on September 3rd and took action both on and off the battlefield. When the Germans invaded Poland in 1939 they responded with acts of war. In 1940, Germany was expanding its power that France had to call an armistice with Germany. This allowed Germany to take over the northern part of France. Grothendieck was still able to receive an education. He just had to hide in the woods whenever authorities were looking for Jews. This also coincided with Grothendieck’s father being handed over to the Nazis by the French Vichy Government, a government that was taken over by the Germans themselves. Even though his family was in an internment camp, Grothendieck was still able to receive an education and become the prominent mathematician people know to this day. 1945 marked the end of World War II with the German’s and Japan’s defeat by the Allies. Grothendieck was able to move out with his mother to the village of Maisargues near the University of Montpellier.

Grothendieck’s “golden age” was around the time of the (1955-1975). The Vietnam War didn’t have much relevance to Grothendieck until towards the end of the war in the late 1960s. Before then, Grothendieck became very successful in mathematics. He established seminars about algebraic geometry which had themes of geometry, number theory, topology, and . In addition, he introduced the theory of schemes, theory of topoi, proved the Riemann-Roch theorem, had a family, and “received” the Fields Medal (the director of IHES took it in his honor). Meanwhile in Vietnam, the northern and southern parts - the communist regime and the Viet Cong - were in government conflict and went to war. It was a devastating war that grew fairly unpopular back at home. The United States intervened and tried to aid the Viet Cong. During this time, Grothendieck declared himself a citizen of the world and visited North Vietnam during the time the United States bombed North Vietnam. While the three nations were at war, Grothendieck lectured at a secret location at Hanoi University in Vietnam for one and a half weeks. He taught those who went to his lectures about general mathematics. Those who tend to stay longer during the time period he stayed learned more about .

After World War II, there were post-war rivalries with the United States and the Soviet Union. It spanned all the way to 1991 and was known as the Cold War. This wasn’t necessarily a war where people fought. Moreover, it was just the United States and Soviet Union threatening each other that they will use weapons of mass destruction (nuclear weapons essentially), if they don’t cooperate with one another or give into another’s needs. It was practically a stalemate with other significant events to get in the way (i.e. Cuban Missile Crisis). Elsewhere, Grothendieck resigned from IHES after discovering the school used military funds to support the school and even abandoned mathematics. During this time, he turned political protest, particularly nuclear proliferation (interesting coincidence).

As the Cold War was coming to an end, Grothendieck retired and wrote a lot of his work in thousands and thousands of pages both math and non-math related. During the official end of the Cold War and the establishment of MAD (Mutually Assured Destruction), just like the war, Grothendieck disappeared. No one had any idea where he went. Overall, Grothendieck was part of the struggles and conflicts during the time he was alive and either related to it in some way or was off with his own success in mathematics.

7 Significant Mathematical Progress During the Alexander Grothendieck’s Lifetime

Grothendieck’s work typically guided others in the right direction in order for those mathematicians to make their own progress. He went very in depth, not only telling how to do it, but explaining to everyone what exactly he is defining and why. There were also many other mathematicians that made progress and discoveries over the course of the 20th century.

Firstly, Andre Weil was a mathematician that Grothendieck actually cooperated with during his studies. Weil was a war refugee from the war in Europe. His theorems, which connected to number theory, algebra, geometry, and topology showed to be one of the greatest achievements in modern mathematics. He also set up the famous Bourbaki group. Grothendieck met him through his professor’s seminar held in Paris and was advised by Weil to go to Nancy to continue his studies and research. Grothendieck’s theory of schemes helped certain amounts of Weil’s number theory conjectures get solved.

Hilbert announced his famous twenty-three problems that really stumped a lot of mathematicians. However, a mathematician named in the 1960s proved Cantor’s hypothesis of the size of infinite sets can be both true and NOT true. He also proved that there were two completely separate mathematical worlds - one containing a continuum hypothesis.

A Russian mathematician named Yuri Matiyasevich proved Hilbert’s tenth problem was impossible. Hilbert’s tenth problem asked if there was a way to determine when polynomial equations have a solution in whole numbers. In other words, Matiyasevich proved there was no distinct pattern to finding whole number solutions in polynomial equations. The irony is that Matiyasevich built his work off of an American mathematician named Julia Robinson and this was during the time of the Cold War. The fact that they helped each other out was a great example of internationalism during that time.

This was also the time that the chaos theory had been introduced. This theory means that there are some systems that behave randomly even though they’re not random at all; they don’t have a distinct pattern that no one can pick out. However, there are also some systems that can be predictable but the details are hard to pick out.

Edward Lorenz became interested in the chaos theory through a weather prediction. It was accidental dis- covery that occurred in 1961 when a computer saved with three digits instead of the usual six. This small change produced drastically different results. He was the one that developed the “butterfly effect”: that one small change could drastically change the outcome in the distant future. This was demonstrated through his , the smallest unit of distinct basins of attraction, in coordination with oscillator.

In addition to the chaos theory, the four color theorem was proved by Kenneth Appel and Wolfgang Haken using a computer. This was the first time a theory was proven through a computer. The theory stated that “in a given separation of a plane in contiguous regions”, all the regions can be colored at most by four colors without adjacent regions being the same color. This theory took 1,200 hours at the computer analyzing over 1,500 configurations. Pictured below is an example proving the theorem:

8 Andrew Wiles, a British mathematician in the late twentieth century proved Fermat’s Last Theorem for ALL numbers. In reality though, he didn’t do it on his own. He got help from previous mathematicians to come up with the final proof of Fermat’s Last Theorem. These include Goro Shimura, Yutaka Taniyama, Gerhard Frey, Jean-Pierre Serre and to aid Wiles over the years. The proof is said to be over 100 pages long. To note: Fermat’s Last Theorem is: an + bn = cn when n > 2

Another famous mathematician is John Nash who established important work in game theory, which is the analysis of strategies when dealing with competitive situations, differential geometry, and partial differential equations. His work contributed to complex systems in daily life such as market economics, computing, artificial intelligence, accounting, and military theory. John Nash battled paranoid schizophrenia and became popular in the award winning film “A Beautiful Mind”.

Overall, there are many mathematicians who contributed during Grothendieck’s life. The twentieth century was a time where mathematics exploded and many scholars started to making discoveries and theories as well as proving theories that were developed in the past.

Connections Between History and the Development of Mathematics

His time era was when mathematics was continuing to expand greatly. It continued the trend from the 19th century of generalization and abstraction. During this time, mathematics was also transforming from leisure time to an actual profession. There was thousands of PH.D students involved every year as well as jobs in teaching and in the industry of mathematics. In fact, the development of all the theories and proofs has created significant inventions and ideas that we hold today.

Perhaps the one of the most significant between history and the development of mathematics would have to be Alan Turing. Turing at the time was one of the leading mathematicians in the world and was called along with a few others to help win World War II by breaking German enigma. His methods were generally confusing but in the end, he created a breakthrough and basically developed the first computer. His mathematical work led to the invention of computers and artificial intelligence, two grand concepts that society uses a lot in today’s world.

In a general sense, the theory of topoi developed by Grothedieck himself are referred to and highly relevant to today. Basically the theory of topoi is the combination of the “concepts of sheaf and closure under categorical operations”. For those who don’t know, a sheaf tracks data attached to open sets of the topological space. This theory plays a role in cohomology - an invariant of topological space, detecting the “holes” in space. This, along with other theories and discoveries he created, developed a new way to look at mathematics and was able to help other mathematicians solve the answer to very difficult problems.

The development of mathematics also contributed to many known games that society still plays today. During this time, mathematics contributed to a concept called game theory. A lot of mathematicians contributed to it. One example is John Horton Conway, a mathematician who established the rules of the “Game of Life” in 1970. These rules showed an example of “cellular automaton” meaning that there’s a pattern of cells that evolve and grow in a grid. Mathematics continued to grow even more as famous puzzles started to appeal to the general public. Erno Rubik, a professor in Budapest, created the famous and strategic Rubik Cube in 1974. A mathematician named Leonhard Euler is mainly responsible for another well known math game we know today as Sudoku in 1980.

Computers were starting to develop even more after Turing’s invention. Mathematicians started using these computers for research. In 1952, the early computer identified one of Mersenne’s prime numbers. A Mersenne

9 prime number is one less than a power of two: p = 2n − 1 The one they found was 2257 − 1 . This was the thirteenth Mersenne prime number where the last one was found seventy-five years ago. In fact, as the internet was coming into play, in 1990, a group called the Great Internet Mersenne Prime Search (GIMPS) is a collaborative group dedicated to finding these numbers. This group is still around today searching for the next Mersenne prime number.

Grothendieck wasn’t the only key figure in mathematics during the twentieth century. Those he was known as one of the more dominate mathematicians of his era, other mathematicians made discoveries that connected to different historic events worldwide.

Remarks

Grothendieck had a very interesting personality. Since he was a kid, he would tend to stay isolated from the rest of society and would focus a lot on mathematics. Despite how intelligent he is on the topic, he tends to put his morals over math no matter what. Some examples include the Fields Medal in 1966. Because of his values of he holds on pacifism and campaigning against military build-up in the 1960s, he didn’t even go all the way to Moscow to receive the honor. Also, because of his values, he left IHES because they were being funded by military. He admitted it to be “spiritual stagnation”. He even disappeared all of a sudden in 1991 and no one knew where or when he left. Despite all these things, Grothendieck is still known as one of the most dominates mathematicians in the twentieth century because he was able to solve many unknown theories and problems other people had. He gave clear and concise definitions of geometric, algebraic, and topological terms. He cooperated with some of the leading mathematicians as well and was able to pass them to make his own discoveries.

Overall, though Grothendieck’s personality may be a little off, he never fell short to mathematics. He always went above and beyond the others.

References

1. http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se- faire-oublier 1142614

2. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Grothendieck.html

3. http://www.telegraph.co.uk/news/obituaries/11231703/Alexander-Grothendieck-obituary.html

4. http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Cartan Henri.html

5. http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Chevalley.html

6. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Delsarte.html

7. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Dieudonne.html

8. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Schwartz.html

9. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Weil.html

10. http://www.history.com/topics/world-war-ii

11. http://www.ushmm.org/wlc/en/article.php?ModuleId=10005137

10 12. http://www.britannica.com/event/Vietnam-War

13. http://www.historylearningsite.co.uk/modern-world-history-1918-to-1980/the-cold-war/what-was-the-cold- war/

14. http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Lebesgue.html

15. http://www.math.umn.edu/ garrett/m/fun/Notes/01 spaces fcns.pdf

16. https://proofwiki.org/wiki/Definition:Fr%C3%A9chet Space %28Topology%29

17. http://www2.stetson.edu/ efriedma/periodictable/html/Uuo.html

18. http://www.storyofmathematics.com/20th.html

19. http://mathworld.wolfram.com/Attractor.html

20. http://plato.stanford.edu/entries/category-theory/

21. https://www.rubiks.com/about/the-history-of-the-rubiks-cube

22. http://www.math.uchicago.edu/ may/VIGRE/VIGRE2009/REUPapers/Talovikova.pdf

23. http://www.sudokudragon.com/sudokuhistory.htm

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