Pierre Deligne

Pierre Deligne, March 2005

Pierre René, Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian mathematician. He is known for fundamental work on the Weil conjectures, leading finally to a complete proof in 1973.

Life

He was born in Brussels, and studied at the Universite Libre de Bruxelles (ULB).

After completing a doctorate under the supervision of Alexander Grothendieck, he worked with him at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 1968, he also worked with Jean-Pierre Serre; their work led to important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions. Deligne's also focused on topics in Hodge Theory. He introduced weights and tested them on objects in complex geometry. He also collaborated with David Mumford on a new description of the moduli spaces for curves. Their work came be seen as a brilliant introduction to algebraic stacks, and recently has been applied to questions arising from string theory. Perhaps Deligne's most famous contribution was his proof of the third and last of the Weil conjectures. As a corollary he proved that the celebrated Ramanujan-Peterrson conjecture on modular forms.

From 1970 until 1984, when he moved to the Institute for Advanced Study in Princeton, Deligne was a permanent member of the IHÉS staff. During this time he did much important work outside of his work on algebraic geometry. In joint work with George Lusztig, Deligne and Lusztig applied étale cohomology to construct representations of finite groups of Lie type; with Rapoport, Deligne worked on the moduli spaces from the 'fine' arithmetic point of view, with application to modular forms. He received a Fields Medal in 1978.

In terms of the completion of some of the underlying Grothendieck program of research, he defined absolute Hodge cycles, as a surrogate for the missing and still largely conjectural theory of motives. This idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. He reworked the tannakian category theory in his paper for the Grothendieck Festschrift, employing Beck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimate Weil cohomology. All this is part of the yoga of weights, uniting Hodge theory and the l-adic Galois representations. The Shimura variety theory is related, by the idea that such varieties should parametrize not just good (arithmetically interesting) families of Hodge structures, but actual motives. This theory isn't yet a finished product – and more recent trends have used K-theory approaches.

He was awarded the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004 and the Wolf Prize in 2008.

In 2006 he was ennobled by the Belgian king as viscount.^[1]

In 2009, Deligne was elected a foreign member of the Royal Swedish Academy of Sciences.^[2]

Selected publications

• Deligne, Pierre (1974). "La conjecture de Weil: I". Publications Mathématiques de l'IHÉS 43: 273–307. http://www.numdam.org/item?id=PMIHES_1974__43__273_0. 
• Deligne, Pierre (1980). "La conjecture de Weil: II". Publications Mathématiques de l'IHÉS 52: 137–252. http://www.numdam.org/item?id=PMIHES_1980__52__137_0. 
• Deligne, Pierre; Mostow, G. Daniel (1993). Commensurabilities among Lattices in PU(1,n). Princeton, N.J.: Princeton University Press. ISBN 0691000964. 

See also

• Deligne conjecture
• Deligne-Mumford moduli space of curves
• Deligne cohomology

References

External links

• O'Connor, John J.; Robertson, Edmund F., "Pierre Deligne", MacTutor History of Mathematics archive, http://www-history.mcs.st-andrews.ac.uk/Biographies/Deligne.html .
• Pierre Deligne at the Mathematics Genealogy Project