Belgium/USA

Pierre Deligne

2004 Balzan Prize for Mathematics

For major contributions to several important domains of mathematics (including algebraic geometry, algebraic and analytic number theory, group theory, topology, and 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).

Pierre Deligne became famous in the mathematical world at an early age through his brilliant proof of the “Weil conjectures”, which concern the number of solutions of systems of polynomial congruences (including the so-called “Riemann conjecture over finite fields”). These conjectures were both exceptionally hard to solve (leading specialists, including A. Grothendieck, had worked on them) and most interesting in view of the far-reaching consequences stemming from their solution. The proof, subject matter of two celebrated papers totalling some 150 pages in the “Publ. Math. I.H.E.S.” (1974 and 1980) was to make use in a remarkably ingenious way of a large combination of very difficult techniques; a real tour de force. It earned its author the Fields Medal in 1978. This first achievement of Pierre Deligne was followed by several others of similar importance. They all demonstrate difficulty in technique and inventiveness in method, as well as extreme variety in type. 

As for the results themselves, some are “elementary”, in that the main statements can be understood by almost any professional mathematician. For instance: the irreducibility of the space of curves of given genus (an early joint paper with D. Mumford, 1969); the definition and application of “buildings” of generalized braid groups (1972); a new solution (also in the early Seventies) of Hilbert’s 21st problem; an epoch-making paper written in common with G. Lusztig on linear representations of finite simple groups of Lie type (“Annals of Mathematics”, 1976); the construction of a remarkable central extension of the group of rational points of a reductive group over a field F by the group K₂ (F) (a construction first described in an unpublished seminar in 1977-1978 and further investigated in a 1996 paper in the “Publ. Math. I.H.E.S.”); the study with G. Mostow of the monodromy of hypergeometric functions (1986). 
Other results are more technical but equally profound, creating new and powerful tools; let us just mention a few titles: La théorie de Hodge II and III (two fundamental papers in the “Publ. Math. I.H.E.S.”, 1971 and 1974; nr. I was just an announcement); Le symbole modéré (ibid. 1991); Faisceaux pervers (in “Astérisque”, vol. 100, 1983, 5-171, joint work with A.A. Beilinson and J. Bernstein); Catégories tannakiennes (in The Grothendieck Festschrift, vol. II, 1990); à quoi servent les motifs? (in Motives, “Proc. Symp. Pure Math.”, 55, 1, A.M.S., 1994; motives are a conjectural notion, created by A. Grothendieck in the late Sixties, rich in implications and often exemplified by Pierre Deligne). 
A remarkable feature of Pierre Deligne’s thinking is that, when confronted with a new problem or a new theory, he understands and, masters 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 discussion. This flexibility is one of the reasons for the universality of his mathematical work. 

Pierre Deligne has written about a hundred papers, (including works of collaboration) most of them quite lengthy. 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 down!), the volume of his publications is a true measure of the richness of his scientific production.