# Pierre René Deligne

### Quick Info

Born
3 October 1944
Etterbeek, Brussels, Belgium

Summary
Pierre Deligne is a Belgian mathematician who won a Fields Medal for his work on algebraic number theory.

### Biography

Pierre Deligne was born in Etterbeek, one of the nineteen suburban districts that, together with central Brussels, make up Greater Brussels. He attended primary school at Schaerbeek, another of the nineteen suburban districts northeast of central Brussels, from September 1950 to June 1956. In September of the same year he began his secondary schooling at the Athénée Adolphe Max in Brussels. He graduated from secondary school in June 1962 and entered the Free University of Brussels in September.

Although Deligne was an undergraduate at the Free University of Brussels from 1962 to 1966, he spent the academic year 1965-66 at the École Normale Supérieure in Paris. He received his Licence en mathématiques in November 1966, the equivalent of a B.A. He continued to study for his doctorate at the Free University of Brussels and in September 1967 he was a junior scientist at the Fond National de la Recherche Scientifique in Brussels, at the same time being a guest at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette in France where he worked with Alexandre Grothendieck. He was awarded his Doctorat en mathématiques by the Free University of Brussels in November 1968.

After the award of his doctorate, Deligne went to the Institut des Hautes études Scientifiques at Bures-sur-Yvette in France where he was a visiting member until February 1970 when he became a permanent member of the Institute. At the IHES he worked with Grothendieck:
... initially on the generalisation of Zariski's main theorem. He also worked closely with Jean-Pierre Serre, leading to important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions. He also collaborated with David Mumford on a new description of the moduli spaces for curves: this work has been much used in later developments arising from string theory.
Deligne remained based at the Institut des Hautes Études Scientifiques until 1984 when he went to the Institute for Advanced Study at Princeton in the United States, where he was appointed a professor.

André Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946). This used Zariski's ideas and also made good use of geometric concepts. Weil's work on polynomial equations led to questions on what properties of a geometric object can be determined purely algebraically. Weil's work related questions about integer solutions to polynomial equations to questions in algebraic geometry. He conjectured results about the number of solutions to polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation. The third of his conjectures was a generalisation of the Riemann hypothesis on the zeta function. These problems quickly became major research challenges to mathematicians.

A solution of the three Weil conjectures was given by Deligne in 1974. This work brought together algebraic geometry and algebraic number theory and it led to Deligne being awarded a Fields Medal at the International Congress of Mathematicians in Helsinki in 1978. A solution to these problems required the development of a new kind of algebraic topology. Tits said:
These conjectures were both exceptionally hard to settle (the best specialists, including A Grothendieck, had worked on them) and most interesting in view of the far-reaching consequences of their solution.
Deligne has worked on many other important problems. The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, $L$-series and the Langlands conjectures, and representations of algebraic groups.

In addition to the Fields Medal, Deligne was awarded the Crafoord Prize of the Royal Swedish Academy of Sciences in 1988:-
... for his fundamental research in algebraic geometry.
Deligne has been awarded many other honours for his outstanding contributions. For example he was awarded the Francois Deruyts prize by the Royal Belgium Academy of Science in June 1974, the Henri Poincaré medal by the Paris Academy of Sciences in December 1974, and the Doctor A De Leeuw-Damry-Bourlart Prize by the Fond National de la Recherche Scientifique in 1975. He has received honorary doctorates from the Flemish University of Brussels in 1989, and from the École Normale Supérieure in 1995. He has been elected a member of the Paris Academy of Sciences in 1978 and by the American Academy of Arts and Sciences in the same year.

In 2004 Deligne was elected an honorary member of the London Mathematical Society [2]:-
... in recognition of his monumental contributions to algebraic geometry.

Viewed as a whole, Deligne's work concerns many different aspects of the cohomology of algebraic varieties. It has turned Grothendieck's philosophy of motives from a conjectural program into what is the driving force behind many of the most subtle areas of current algebraic geometry and arithmetic. Through an unparalleled blend of penetrating insights, fearless technical mastery and dazzling ingenuity, Deligne has singlehandedly brought about a new understanding of the cohomology of varieties, both classical and in finite characteristic, with numerous applications to deep problems in geometry and number theory.
Very recently Deligne received the 2004 Balzan Prize in Mathemtics awarded by the International Balzan Foundation:-
... for major contributions to several important domains of mathematics (like algebraic geometry, algebraic and analytic number theory, group theory, topology, Grothendieck theory of motives), enriching them with new and powerful tools and with magnificent results such as his spectacular proof of the "Riemann hypothesis over finite fields" (Weil conjectures).
Jacques Tits, as a member of the Balzan Prize committee, announced the prize on 7 September 2004 in Milan. He described Deligne's work, then ended by making the following comments:-
A remarkable feature of Pierre Deligne's thinking is that, when confronted with a new problem or a new theory, he understands and, so to speak, makes his own its basic principles at a tremendous speed, and is immediately able to discuss the problem or use the theory as a completely familiar object. Thus, he readily adopts the language of the people he is talking to when engaged in discussions. This flexibility is one of the reasons for the universality of his mathematical work.

Alone or in collaboration, Pierre Deligne has written about a hundred papers, most of them of sizeable length. Because of the conciseness of his style and of his habit of never writing the same thing twice (in fact, quite a few of his best ideas have never been written!), the volume of his publications is a true measure of the richness of his scientific production.
As winner of the Balzan Prize, Deligne received 1 million Swiss francs (about US\$800,000), half of which would go to research projects involving young researchers in his field. The prize ceremony took place on 18 November 2004 in Rome.

### References (show)

1. N M Katz, The work of Pierre Deligne, Proceedings of the International Congress of Mathematicians, Helsinki 1978 (Helsinki, 1980), 47-52.
2. Honorary member : Pierre Deligne, Bull. London Math. Soc. 36 (2004), 855-856.
3. R Kiehl, Zum mathematischen Werk von Pierre Deligne, Jahrbuch Überblicke Mathematik, 1979 (Mannheim, 1979), 169-172.
4. D Mumford and J Tate, Fields medals. IV. An instinct for the key idea, Science 202 (4369) (1978), 737-739.
5. T Oda, Works of P Deligne I (Japanese), Sugaku 31 (1) (1979), 18-25.
6. H Yoshida and K Aomoto, Works of P Deligne II (Japanese), Sugaku 31 (1) (1979), 25-29.