Pierre Deligné is a research mathematician who has excelled at making connections between various fields of mathematics. His research has led to several important discoveries, the most critical of which is the proof of three famous conjectures made by the mathematician André Weil. For this work, Deligné received both the Fields Medal, the highest honor in mathematics, and the Crafoord Prize. In recognition of his reception of the Fields Medal, David Mumford and John Tate, both of the Harvard University Department of Mathematics, wrote in Science magazine that "There are few [mathematical] subjects that [Deligné's] questions and comments do not clarify, for he combines powerful technique, broad knowledge, daring imagination, and unfailing instinct for the key idea."
Pierre René Deligné was born on October 3, 1944, in Brussels, Belgium, where he and his parents, Albert and Renee Bodart Deligné, lived throughout his childhood. The young Deligné showed an early affinity for mathematics, and his interest was encouraged by M. J. Nijs, his high school teacher. Nijs loaned Deligné several books by Nicolas Bourbaki that introduced concepts of modern mathematics, such as topology, long before discussing the topics traditionally studied first. Despite the unfamiliar and complicated terminology, Deligné's understanding of mathematics flourished, and after completing high school he enrolled at the University of Brussels. Deligné obtained his degree in mathematics there in 1966 and remained for graduate study.
Deligné's adviser at the University of Brussels, group theorist Jaques Tits, suggested in 1965 that Deligné travel to Paris. Since Deligné was interested in algebraic geometry, Tits felt that he should study where some of the most important researchers in that field were teaching and researching--mathematicians such as Jean-Pierre Serre and Alexander Grothendieck. Deligné went and met both Serre and Grothendieck; his association with them would strongly influence his career. After returning to Brussels to complete work on his dissertation, he received his Ph.D. in 1968.
Following completion of his doctorate, Deligné took up residence in Bures-sur-Yvette, a small community south of Paris, where the Institut des Hautes Etudes Scientifiques (Institute for Advanced Scientific Study--IHES) is located. He had been appointed a visiting member of this organization so that he could continue his research with Grothendieck; he became a permanent member of the IHES in 1970. For several years, Grothendieck had been working to generalize and update the field of algebraic geometry by making it more compatible with recent abstract mathematical theories. Deligné admired and learned from Grothendieck's work, although he followed a different approach. Whereas Grothendieck tried to connect algebraic geometry with all other fields by creating new theories or rules, Deligné instead worked to uncover connections already implied by previous work in these fields. Contrasting the two men's styles, Mumford and Tate observed, "One could say that Grothendieck liked to cross a valley by filling it in, Deligné by building a suspension bridge."
Conquers the Weil Conjectures
A prime example of Deligné's methods is his work on the Weil conjectures. Proposed in 1949 by the mathematician André Weil, these three conjectures state that it should be possible to determine the number of solutions for certain systems of equations by predicting the shapes of the graphs of the solutions. In other words, by using certain topological concepts, algebraic results can be obtained. As explained by Michael Atiyah in his 1975 Bakerian Lecture, this amounted to finding an algebraic technique for identifying holes in the manifold of complex solutions of an equation. Although Weil felt certain that he was correct, he was never able to prove his conjectures. Over a period of several years, Deligné whittled away at the conjectures. Combining the new theory of étale cohomology (a branch of topology), which had been developed by Grothendieck, and a related conjecture by the Indian number theoristS. I. Ramanujan, he completed the final proof in 1973.
Deligné's work has been valued not only because he solved an important problem in mathematics, but also because he proved that seemingly disparate subjects can be connected. Referring to Deligné's use of a 1939 paper on Ramanujan's conjecture along with the new étale cohomology, Mumford and Tate wrote, "It is hard to imagine two mathematical schools more different in spirit and outlook than were those of the British analytic number theorists in the 1930s and of the French algebraic geometers in the 1960s. That Deligné's proof is a blend of ideas from both is an indication of the universality of his mathematical taste and understanding." For this reason, as much as for actually proving the Weil conjectures, the International Mathematics Union in 1978 awarded Deligné its highest honor, the Fields Medal, noting that his work "did much to unify algebraic geometry and algebraic number theory."
Continues to Develop Algebraic Geometry
Deligné continued to study the Weil conjectures even after his initial success, attempting to use automorphic forms (equations involving multiple functions) and prime numbers to determine more and more exact solutions. He worked on several problems proposed by the American mathematician Robert Langlands, who was leading a major research program in the area of automorphic forms at Princeton's Institute for Advanced Studies (IAS). At Langlands' invitation, Deligné traveled to the United States in 1977 to help organize a conference at Oregon State University.
Also, in the late 1970s, Deligné gave a series of lectures in étale cohomology with the help of Grothendieck and others in the field of algebraic geometry. These lectures were considered definitive in describing this relatively new field, but Deligné's contributions were not limited to lecturing. He added significantly to the content by his work with Shimura varieties and by his proofs of some conjectures proposed by William V. D. Hodge. In 1988, the Royal Swedish Academy of Sciences awarded Deligné and Grothendieck the Crafoord Prize for Mathematics for their work in defining étale cohomology and applying it to algebraic geometry (Grothendieck declined his prize).
In 1980, Deligné married Elena Vladimirovna Alexeeva, who would become the mother of his two children. He enjoys simple pleasures such as vegetable gardening, bicycle riding, and hiking. He brought his young family to the United States in 1984 to continue his mathematical research at the IAS, where he has remained. In 1993, Deligné and G. Daniel Mostow coauthored a book titled Commensurabilities Among Lattices in PU(1,n). Reviewing this book for the Bulletin of the American Mathematical Society, P. Beazely Cohen and F. Hirzebruch wrote that the authors "extract the best aspects of the previous techniques of algebraic and differential geometry . . . together with function theory, giving an overall coherent presentation yielding new results." A quarter of a century after cracking the Weil conjectures, Deligné has not lost his powerful touch for creating mathematics.
Recent Updates
November 18, 2004: Deligne was awarded the 2004 Balzan Prize in mathematics at a ceremony in Rome, Italy. The prize carries a cash award of one million Swiss francs (about $865,000), half of which must be used for projects involving young researchers. The Balzan Prizes are administered by the International Balzan Foundation and are aimed at fostering worldwide achievement in the sciences and humanities. Source: Balzan Foundation, www.balzan.com, November 30, 2004.
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