About André Abraham Weil (Weill)
André Weil (6 May 1906 – 6 August 1998) was one of the century's most influential pure mathematicians, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. The philosopher Simone Weil was his sister.
André Weil was born in Paris, the son of Jewish parents. His mother Selma came from a family of Austrian Jews, while his father, Bernard Weil, was a medical doctor, and his grandfather, Abraham Weil was a prominent member of the Jewish community in Alsace. Hower, it was not till 1919 André and his sister, Simone, learnt for the first time that they were Jewish. As André writes, this was clearly quite a traumatic event. That their parents had been able to keep it from them until this time is an indication of how well integrated the Jewish community was in French society.
André fell in love with mathematics at an early age, and he writes that by the age of ten he was passionately addicted to it. Today, André Weil is especially known for his foundational work in number theory and algebraic geometry. He was a founding member and the de facto early leader of the influential Bourbaki group.
His "Weil conjectures" were very influential; these and other works laid the groundwork for many of Grothendieck's achievements. Weil proved a special case of the Riemann hypothesis; he contributed, at least indirectly, to the recent proof of "Fermat's last Theorem;" he also worked in group theory, general and algebraic topology, differential geometry, sheaf theory, representation theory, and theta functions. He invented several new concepts including vector bundles, and uniform space. His work has found applications in particle physics and string theory. He is considered to be one of the most influential of modern mathematicians.
There were other things of importance in his life as well as mathematics, however, for he loved to travel. By the age of sixteen he had read the Bhagavad Gita in the original Sanskrit.
Weil studied at the École Normale in Paris, and after graduating he spent the summer vacation walking in the French Alps, always taking a notebook with him in which he made his mathematical calculations. At this time he was particularly fascinated by solving Diophantine equations. After the summer vacation he went to Rome and then on to Göttingen where he produced his first substantial piece of mathematical research on the theory of algebraic curves. He then undertook research for his doctorate in the University of Paris, supervised by Hadamard. He developed for his thesis the ideas on the theory of algebraic curves which he had begun to study at Göttingen. However, Hadamard wanted his brilliant student to aim higher and try to prove the Mordell Conjecture. Weil chose not to follow his supervisor's advice. He wrote later:-
My decision was a wise one: it was to take more than half a century to prove Mordell's Conjecture.
He received his D.Sc. from Paris in 1928. He then taught at different universities, for example the Aligarh Muslim University in India from 1930 to 1932. He had first discussed with Syed Masood, the Minister of Education for Hyderabad, obtaining an appointment to a chair in French Civilization at Aligarh University but, despite the promise, he received a telegram from Syed Masood:-
Impossible to create chair of French civilisation. Mathematics chair open.
He also worked at the University of Strasbourg, France, from 1933 until the outbreak of World War II. It was here that he became involved with the famous group of mathematicians writing under the name Nicolas Bourbaki. We give more details of this collaboration below.
The war was a disaster for Weil who was a conscientious objector and so wished to avoid military service. He fled to Finland, to visit Rolf Nevanlinna, as soon as war was declared. This was an attempt to avoid being forced into the army, but it was not a simple matter to escape from the war in Europe at this time. Weil was arrested in Finland and when letters in Russian were found in his room (they were actually from Pontryagin describing mathematical research) things looked pretty black. One day Nevanlinna was told that they were about to execute Weil as a spy, and he was able to persuade the authorities to deport Weil instead.
From Finland he was sent back to France where he was put in prison. Weil was certainly in great danger at this time, partly because he was Jewish, partly because he had a sister Simone Weil who was a mystic philosopher and a leading figure in the French Resistance. The dangers of his predicament made Weil decide that being in the army was a better bet and he was able to argue successfully for his release on the condition that indeed he did join the army.
Having used the army as a reason to get out of prison, Weil had no intention of serving any longer than he possibly could. As soon as the chance to escape to the United States came, he took it at once. In the United States he went to Pennsylvania where he taught from 1941 at Haverford College and at Swarthmore College. In 1945 he accepted a position in São Paulo University, Brazil, where he remained until 1947. In 1947 Weil returned to the United States and he was appointed to the faculty of the University of Chicago, a position he continued to hold until 1958. From 1958 he worked at the Institute for Advanced Study at Princeton University. He retired in 1976, becoming Professor Emeritus at that time.
Weil's research was in number theory, algebraic geometry and group theory. His work is summarised in :-
Beginning in the 1940s, Weil started the rapid advance of algebraic geometry and number theory by laying the foundations for abstract algebraic geometry and the modern theory of abelian varieties. His work on algebraic curves has influenced a wide variety of areas, including some outside mathematics, such as elementary particle physics and string theory.
In fact Weil's work in this area was basic to work by mathematicians such as Yau who was awarded a Fields Medal in 1982 for work in three dimensional algebraic geometry which has major applications to quantum field theory.
Yau is not the only mathematician who received a Fields Medal for work which continued that begun by Weil. In 1978 Deligne was awarded a Fields Medal for solving the Weil Conjectures. Again we quote  for a description of Weil's fundamental contribution:-
One of Weil's major achievements was his proof of the Riemann hypothesis for the congruence zeta functions of algebraic function fields. In 1949 he raised certain conjectures about the congruence zeta function of algebraic varieties over finite fields. These Weil conjectures, as they came to be called, grew out of his deep insight into the topology of algebraic varieties and provided guiding principles for subsequent developments in the field.
Weil's work on bringing together number theory and algebraic geometry was highly fruitful. The foundations of many topics studied in depth today were laid by Weil in this work, such as the foundations of the theory of modular forms, automorphic functions and automorphic representations.
However, Weil's work was of major importance in a number of other new mathematical topics. He contributed substantially to topology, differential geometry and complex analytic geometry. It was not just to these areas that he contributed but, even more importantly, his work brought out fundamental relationships between the areas when he studied harmonic analysis on topological groups and characteristic classes. Also bringing these areas together was his work on the geometric theory of the theta function and Kähler geometry.
Together with Dieudonné and others, Weil wrote under the name Nicolas Bourbaki, a project they began in the 1930s, in which they attempted to give a unified description of mathematics. The purpose was to reverse a trend which they disliked, namely that of a lack of rigour in mathematics. The influence of Bourbaki has been great over many years but it is now less important since it has basically succeeded in its aim of promoting rigour and abstraction.
Weil's most famous books include Foundations of Algebraic Geometry (1946) and Elliptic Functions According to Eisenstein and Kronecker (1976).
Weil received many honours for his outstanding mathematics. Among these has been honorary membership of the London Mathematical Society in 1959 and election to a Fellowship of the Royal Society of London in 1966. In addition he has been elected to the Academy of Sciences in Paris and to the National Academy of Sciences in the United States.
Weil was an invited speaker at the International Congress of Mathematicians in 1950 at Harvard and again at the following International Congress in 1954. In 1979 Weil was awarded the Wolf Prize and, in the following year, the American Mathematical Society awarded him their Steele Prize. In 1994 he received the Kyoto Prize from the Inamori Foundation of Japan:-
... for outstanding achievement and creativity.
The citation for the Kyoto Prize reads:-
The results achieved and problems raised by André Weil through his deep understanding of and sharp insight into mathematical sciences in general will continue to have immeasurable influence on the development of mathematical sciences, and to contribute greatly to the development of science, as well as the deepening and uplifting of the human spirit.
He is described as follows in :-
Andre Weil will be remembered for his fundamental work on the frontiers of mathematics, and for his carefully cultivated image as a cantankerous character - belied by his dry sense of humour. The only honour listed in his official biography is "Member, Poldavian Academy of Science and Letters".
(Poldavia was invented home country of the fictional Bourbaki.)
Article by: J J O'Connor and E F Robertson
The Weil family
It is a cold winter day and two children are riding on a Paris bus. The boy is the elder of the two and his little girl sister looks desperately thin and undernourished. Both have bare feet and their teeth are chattering with cold. They turn to the passengers in the bus and explain that their parents neglect them and won't even buy them socks and shoes. The passengers are horrified - but all is not as it seems. In fact the children are André and Simone Weil and they are two of the most intelligent children in the whole of France. They are playing out a game as a way of reacting against their overprotective parents. It is only a small rebellion, for otherwise they conform to a childhood of rigorous intellectual development. Toys are banned at home and instead of the chatter of children they communicate in ancient Greek or rhyming couplets -
The young Weil's conversations though never meant to exclude anyone, were so laced with literary and philosophical allusions that they were barely accessible to outsiders.
Their parents are Bernard and Selma Weil. Salomea Reinherz, or Selma as she called herself, was the daughter of a successful Jewish businessman who dealt in imports and exports. She was born in Russia, but life there became hard after Russian Jews were attacked after the assassination of Tsar Alexander II in 1881. The fact that the assassination was not by a Jew was not relevant as rumours circulated. In these difficult and dangerous times the Reinherz family decided to leave Russia and they emigrated to Belgium in the 1880s. Selma's father was an artistic man who wrote Hebrew poetry. Her mother was a gifted pianist who lived with Bernard and Selma Weil in their Paris home after their marriage. When Selma was young she had wanted to become a medical doctor but her father prevented her going to medical school. The frustration of being prevented from achieving her goals in her own career meant that she wanted desperately for her own children to have the best education and chance that anyone had ever had. Selma was highly intelligent, full of life, and dominated life in the home.
Bernard came from a family of Jewish merchants and several generations of his family had lived in Strasbourg. Although Jewish by race, he was not Jewish by religion. In fact he was against all religion and disliked speaking of his Jewish background. Bernard's father, however, was Abraham Weil, a prominent member of the Jewish community in Alsace while his mother was a deeply religious woman who was horrified at her son's lack of religion. Although she did not live with Bernard and Selma Weil after their marriage, she did live in Paris until her death in the 1930s. Having her living close by certainly made problems for the Weil family since when she visited her son and daughter-in-law, much to her horror, they prepared food for her which was not kosher. Bernard Weil was a highly successful medical doctor who was :-
... kind, loving, and thoroughly enlightened, but taciturn and easily overwhelmed by his forceful spouse. [He] was far too busy with his medical practice, and let his wife make the major decisions concerning their children's education.
André was born on 6 May 1906 and Simone was born on 3 February 1909. The children were so brilliant and Selma wanted such high quality education for them that they attended around seven different schools in the space of five years. The Weils also employed several private tutors as Selma put every ounce of her energy into achieving her educational goals for the children. When she complained to one of André's teachers that he may not have a very good grounding in arithmetic, the teacher replied :-
No matter what I tell him on that subject, he seems to know it already.
André knew that he wanted to be a mathematician from the age of eight after he found a mathematics book in his aunt's house and studied it intently. From that time on he displayed a passion for mathematics which at first his mother discouraged, but after she saw how impossible it was to stop him thinking about mathematical problems she relented. When André was eight years old he decided that he would teach his five year old sister Simone to read as a birthday present to his father. In only a few weeks he had taught her to read and spell. Simone adored her brilliant older brother and put in a huge effort to please him. By age twelve André had taught himself ancient Greek and Sanskrit. Also by this time he was an excellent violinist. Simone learnt ancient Greek and a number of modern languages in her early teens.
Selma not only had an obsession regarding her children's education but she also had an obsession about cleanliness. She had, as André later wrote :-
... a dread of germs that [she] would carry to an extreme.
Only close family members were allowed to kiss the children and by the time Simone reached the age of four even her parents didn't kiss her. The children had to wash their hands frequently and after doing so would have to open the door of the dining room with their elbows so that they touched nothing before eating their food. Selma's obsession seemed to transfer itself to Simone who hardly eat any food at all. As a child she would cry if anyone touched her and in later life she could not bare physical contact.
Bernard and Selma Weil lived in a Paris apartment on the Rue de Strasbourg, south of the Gare de l'Est when the children were very young. In 1914 they moved to a larger home on the Boulevard Saint-Michel. However, 1914 saw the start of World War I and Bernard Weil was drafted. He joined the army medical corps and was sent to the front but the sight of so many casualties and the poor state of the medical services available to the wounded soldiers led to him suffering from exhaustion and depression. He was sent away from the front to recover his own health. It was during this period that André gave him the birthday present of Simone reading a newspaper to him which we described above. Bernard was posted to Neufchâtel in the Normandy region of northern France, then to Mayenne in the Loire region of northwest France south of Normandy, next to Laval not far from Mayenne and also on the Mayenne river, and from there to Chartres southwest of Paris. In each if these towns the Weils rented a house so that Selma and the children could be near their father. After the war ended the family returned to their home on the Boulevard Saint-Michel in Paris in January 1919.
In 1919 André and Simone learnt for the first time that they were Jewish. As André writes, this was clearly quite a traumatic event. That their parents had been able to keep it from them until this time is an indication of how well integrated the Jewish community was in French society. They had suffered none of the frightening experiences that Selma had had in her youth in Russia. Bernard also had suffered from anti-Semitism when he was younger and living with his parents in Alsace. André by this time was studying at the Lycée Saint-Louis and beginning to receive unique educational opportunities. In 1920, at the age of fourteen, he first met Hadamard. André later wrote :-
The warmth with which he received me eliminated all distance between us. He seemed to me like a peer, infinitely more knowledgeable, but hardly any older; he needed no effort to make himself accessible to me ...
When André was awarded a school prize he asked Hadamard to help him choose some mathematics books and as a result he became the proud owner of Jordan's Cours d'Analyse and Thomson and Tait's Treatise of Natural Philosophy. He received a special dispensation at age fourteen to take the baccalaureate examination although the official minimum age to take the state examination was seventeen. Although three years below the minimum age, he obtained the highest grades of anyone throughout France. Life at home had continued to be on the highest intellectual level :-
... preferred topics of conversation at the Weils dinner table - music, literature, and André's favourite hobby, the collecting rare editions of Greek and Latin texts - were occasionally held in the family's second languages, German and English. It was a highly cosmopolitan family. Selma Weil, who had inherited a tidy income from her prosperous merchant father, loved to travel, and several times a year devised ingenious vacations for the family to enjoy together. In fact one is bound to be struck by the variety of fashionable, luxurious vacations the commanding Selma Weil planned for her family. Spending substantial sums on their travel, the Weils took off, nor only on summer vacations but on any other major holidays - Christmas, Easter, Pentecost, All Soul's Day - to a variety of glamorous destinations, such as biking trips in the Tyrol or hiking in the Black forest.
André spent a year preparing for university entrance after taking the baccalaureate examinations, and during this time he met Sylvain Lévi who was a leading scholar of Indian studies. He began his studies at the école Normale Supérieure which he entered in 1922 and, once there, asked Sylvain Lévi for advice on a book to read in the vacation. Lévi suggested that André read Bhagavad Gita saying :-
Read this. First of all, you cannot understand anything about India if you haven't read it and besides, it is beautiful.
This book had a great influence on André throughout his life. At the école Normale he attended, in addition to mathematics classes, Jules Bloch's course on the Veda, Meillet's lectures on Indo-European linguistics, and Sylvain Lévi's course on Meghaduta which is a poem by Kalidasa, one of the greatest poets from ancient Indian times.
Simone wrote later about how she became depressed because of André's brilliance. One has to remember that these words were written by a girl who was herself brilliant :-
At fourteen I fell into one of those fits of bottomless despair that come with adolescence, and I seriously thought of dying because of the mediocrity of my natural faculties. The exceptional gifts of my brother, who had a childhood and a youth comparable to Pascal's, brought my own inferiority home to me.
Simone attended the Lycée Henri IV while preparing for university. There she was much influenced by émile Chartier who was known by his pen name of Alain. In 1928 she entered the école Normale Supérieure. In May of the following year her parents moved to a new home in Paris, to an apartment on the Rue Auguste Comte. Although by this time André did not live with his parents, they reserved the top floor of the apartment for him. Mrs Reinherz, Selma's mother, who lived with the Weils, died of cancer in 1929 at around the time they moved to their new home.
The biography of André Weil contains details of his life.
Here we briefly note what happened to the other members of Weil's family. Simone became a famous philosopher although she was an anorexic who undertook hard physical labour at times almost to punish herself. She also struggled with becoming a Roman Catholic, but there were certain aspects which she could not accept such as unbaptised infants going to hell. Her violent opposition to Judaism meant that the Old Testament part of Christianity was a huge stumbling block. Simone, together with her parents, fled from the advancing German troops when they invaded France at the start of World War II. Before that she had argued a pacifist policy and she later suffered by seeing herself as part of the reason for the French defeat. They reached Marseilles where they lived until May 1942. There Simone wrote many essays and continued her exploration of Christianity. From Marseilles the three members of the Weil family sailed to Casablanca, sailing from there to New York in June. In November Simone sailed to London leaving her parents in New York. In London she worked for the Free French but became ill and died on 24 August 1943. Basically she died from starvation. Bernard and Selma Weil returned to their apartment on the Rue Auguste Comte in Paris at the end of the war. They then worked hard for the rest of their lives copying out Simone's manuscripts. Bernard died in 1955 and Selma in 1965. André Weil's family owned the apartment on the Rue Auguste Comte until the late 1990s.
References (3 books/articles)
Article by: J J O'Connor and E F Robertson
The apprenticeship of a mathematician By André Weil