One of the most respected mathematicians of the second half of this century, Andre Weil is best known for two things: his fundamental discoveries in number theory, and his membership of the secretive group known as Bourbaki, which redefined the foundations of modern pure mathematics.

He held a professorship at the Institute for Advanced Study in Princeton, New Jersey, where his colleagues included Albert Einstein, J. Robert Oppenheimer, John von Neumann and Kurt Godel. His sister Simone, the renowned mystic and philosopher, active in the French Resistance, died in 1943.

Andre Weil was born in Paris. His father Bernard was a physician; his mother Selma came from a community of Austrian Jews in the Russian port of Rostov-on-Don. By the age of ten he was "passionately addicted" to mathematics, as he says in his autobiography The Apprenticeship of a Mathematician. His other passions were languages -- in 1922, aged 16, he read the Bhagavad Gita in the original Sanskrit -- and travel.

After graduating from the Ecole Normale in Paris he went walking in the French Alps, always taking with him a notebook of mathematical calculations. He was especially fascinated by "Diophantine equations" -- named after the Greek Diophantus, who flourished around AD 250 -- where the name of the game is to find whole numbers that satisfy some stated algebraic relation. (For example: find three whole numbers whose sum is a square and whose sums in pairs are also squares. Diophantus's answer: 41, 80, and 320.)

Weil made his way to Italy, and then to Gottingen, hub of German mathematics and home of the legendary David Hilbert. Here he produced his first serious piece of mathematics -- research on the theory of algebraic curves that answered a problem posed 25 years earlier by the French genius Henri Poincare. This eventually formed Weil's doctoral thesis, but his adviser Jacques Hadamard encouraged him to aim higher, at a related but more difficult problem known as the Mordell Conjecture. Weil decided to ignore this advice, later observing: "My decision was a wise one: it was to take more than half a century to prove Mordell's Conjecture."

His first academic position was in India, at the Aligarh University. Syed Masood, Minister of Education for Hyderabad, had promised him a chair in French civilisation, but the plan went awry. A cable came: "Impossible to create chair of French civilisation. Mathematics chair open." From 1933 to 1939 Weil worked in Strasbourg, where he became involved with the celebrated and now somewhat controversial group known as "Nicolas Bourbaki". The name -- allegedly that of a citizen of the imaginary state of Poldevia -- arose from a spoof lecture presented in 1923 by a practical joker whose real name was Raoul Husson.

The Young Turks who constituted Bourbaki were unhappy with the disorganised state of fundamental mathematical knowledge, and resolved to put it on a sound basis. In so doing they made what many now consider a strategic error: treating every topic in maximum generality, which implied maximum abstraction. This approach rendered Bourbaki's texts incomprehensible to all save the initiated -- but the initiated revelled in them.

The "new math" imported into schools in the 1970s was to some extent modelled on Bourbaki, under the misconception that what was acceptable to trained research mathematicians would also be appropriate for schoolchildren. Bourbaki's influence has now waned, but at the time the group brought some much-needed conceptual clarity to the subject.

In 1939, when war broke out, Weil dodged the draft by visiting Rolf Nevanlinna in Finland. The Finns promptly arrested the suspicious-looking foreigner, and a search of his room brought to light incomprehensible letters in Russian -- actually research news from the distinguished mathematician Lev Pontrjagin. As it happened, Nevanlinna was a reserve colonel in the army.

One day the chief of police casually told him: "Tomorrow we are executing a spy who claims to know you." Nevanlinna asked who, was told it was Weil, and with admirable presence of mind calmly suggested that deportation might be better. The chief of police, who had not thought of this option, agreed that it might be a more sensible course.

By 1941 a complicated series of events had brought Weil to New York, and thereafter the United States became his home. In 1947, in Chicago, he read some old papers by Carl Friedrich Gauss, and these led him to a proof of the so-called Riemann hypothesis for algebraic curves. Generalising wildly -- a rather uncharacteristic act -- he formulated a series of statements that became known as the Weil Conjectures. These turned out to be so important that their proof, nearly thirty years later, earned the young Belgian mathematician Pierre Deligne the Fields Medal -- the mathematicians' equivalent of the Nobel Prize. In 1994 Weil received an equally significant award, the Kyoto Prize, for his conjectures.

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".

His wife Eveline died in 1986. He is survived by two daughters.

© The Times, 1998