Pierre Deligne

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Pierre Deligne, March 2005

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

After completing a doctorate, he worked with Alexander Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalisation within scheme theory of Zariski's main theorem. He 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.

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, besides the proof of the Weil conjectures: in particular with George Lusztig on the use of étale cohomology to construct representations of algebraic groups, and with Rapoport 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.

Deligne has written a book with G. D. Mostow on monodromy. He was awarded the Crafoord Prize in 1988 and the Balzan Prize in 2004.

See also

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

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