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André Weil

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6 May 1906

Paris, France

6 Aug 1998

Princeton, New Jersey, USA

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. 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. 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 Sao 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 .)

Source:School of Mathematics and Statistics University of St Andrews, Scotland