Pierre Deligne
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Pierre Deligne  

Pierre Deligne, March 2005


Born 
Etterbeek, Belgium 
3 October 1944
Nationality  Belgian 
Alma mater  Université libre de Bruxelles 
Known for  Proof of the Weil conjectures Perverse sheaves Concepts named after Deligne 
Awards 
Abel Prize (2013) Wolf Prize (2008) Balzan Prize (2004) Crafoord Prize (1988) Fields Medal (1978) 
Scientific career  
Fields  Mathematics 
Institutions 
Institute for Advanced Study Institut des Hautes Études Scientifiques 
Doctoral advisor  Alexander Grothendieck 
Doctoral students 
Lê Dũng Tráng Miles Reid Michael Rapoport 
Pierre René, Viscount Deligne (French: [dəliɲ]; born 3 October 1944) is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal.
Contents
Biography
Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université 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 JeanPierre Serre; their work led to important results on the ladic representations attached to modular forms, and the conjectural functional equations of Lfunctions. 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 to be seen as an introduction to one form of the theory of 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. This proof completed a programme initiated and largely developed by Alexander Grothendieck. As a corollary he proved the celebrated Ramanujan–Petersson conjecture for modular forms of weight greater than one; weight one was proved in his work with Serre. Deligne's 1974 paper contains the first proof of the Weil conjectures, Deligne's contribution being to supply the estimate of the eigenvalues of Frobenius, considered the geometric analogue of the Riemann hypothesis. Deligne's 1980 paper contains a much more general version of the Riemann hypothesis.
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 applied étale cohomology to construct representations of finite groups of Lie type; with Michael 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 1990 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 ladic 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 is not yet a finished product – and more recent trends have used Ktheory approaches.
Awards
He was awarded the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, the Wolf Prize in 2008, and the Abel Prize in 2013.
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]} He is a member of the Norwegian Academy of Science and Letters.^{[3]}
Selected publications
 Deligne, Pierre (1974). "La conjecture de Weil: I". Publications Mathématiques de l'IHÉS. 43: 273–307. doi:10.1007/bf02684373.
 Deligne, Pierre (1980). "La conjecture de Weil : II". Publications Mathématiques de l'IHÉS. 52: 137–252. doi:10.1007/BF02684780.
 Deligne, Pierre (1990). "Catégories tannakiennes". Grothendieck Festschrift vol II. Progress in Mathematics. 87: 111–195.
 Deligne, Pierre; Mostow, G. Daniel (1993). Commensurabilities among Lattices in PU(1,n). Princeton, N.J.: Princeton University Press. ISBN 0691000964.
 Quantum fields and strings: a course for mathematicians. Vols. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison and Edward Witten. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Vol. 1: xxii+723 pp.; Vol. 2: pp. i–xxiv and 727–1501. ISBN 0821811983.
Handwritten letters
Deligne wrote multiple handwritten letters to other mathematicians in the 1970s. These include
 "Deligne's letter to PiatetskiiShapiro (1973)" (PDF). Retrieved 15 December 2012.
 "Deligne's letter to JeanPierre Serre (around 1974)". 20121215.
 "Deligne's letter to Looijenga (1974)" (PDF). Retrieved 15 December 2012.
Concepts named after Deligne
The following mathematical concepts are named after Deligne:
 Deligne–Lusztig theory
 Deligne–Mumford moduli space of curves
 Deligne–Mumford stacks
 Fourier–Deligne transform
 Deligne cohomology
 Deligne motive^{[4]}
 Deligne tensor product of abelian categories (denoted )^{[5]}
 Langlands–Deligne local constant
Additionally, many different conjectures in mathematics have been called the Deligne conjecture:
 The Deligne conjecture in deformation theory is about the operadic structure on Hochschild cohomology. It was proved by Kontsevich–Soibelman, McClure–Smith and others. It is of importance in relation with string theory.
 The Deligne conjecture on special values of Lfunctions is a formulation of the hope for algebraicity of L(n) where L is an Lfunction and n is an integer in some set depending on L.
 There is a Deligne conjecture on 1motives arising in the theory of motives in algebraic geometry.
 There is a Gross–Deligne conjecture in the theory of complex multiplication.
 There is a Deligne conjecture on monodromy, also known as the weight monodromy conjecture, or purity conjecture for the monodromy filtration.
 There is Deligne conjecture in the representation theory of the exceptional Lie groups.
 There is a Deligne–Langlands conjecture of historical importance in relation with the development of the Langlands philosophy.
 Deligne's conjecture on the Lefschetz trace formula^{[6]} (now called Fujiwara's theorem for equivariant correspondences).^{[7]}
References
 ^ Official announcement ennoblement  Belgian Federal Public Service. 20060718 Archived 30 October 2007 at the Wayback Machine.
 ^ Royal Swedish Academy of Sciences: Many new members elected to the Academy, press release on 12 February 2009^{[dead link]}
 ^ "Gruppe 1: Matematiske fag" (in Norwegian). Norwegian Academy of Science and Letters. Retrieved 26 April 2014.
 ^ motive in nLab
 ^ Deligne tensor product of abelian categories in nLab
 ^ Yakov Varshavsky (2005), "A proof of a generalization of Deligne's conjecture", p. 1.
 ^ Martin Olsson, "Fujiwara's Theorem for Equivariant Correspondences", p. 1.
External links
Wikiquote has quotations related to: Pierre Deligne 
Wikinews has related news: Norwegian Academy of Science and Letters awards Belgian mathematician Pierre Deligne with Abel prize of 2013 
 O'Connor, John J.; Robertson, Edmund F., "Pierre Deligne", MacTutor History of Mathematics archive, University of St Andrews.
 Pierre Deligne at the Mathematics Genealogy Project
 Roberts, Siobhan (20120619). "Simons Foundation: Pierre Deligne". Simons Foundation. — Biography and extended video interview.
 Pierre Deligne's home page at Institute for Advanced Study
 Katz, Nick (June 1980), "The Work Of Pierre Deligne", Proceedings of the International Congress of Mathematicians, Helsinki 1978 (PDF), Helsinki, pp. 47–52, ISBN 9514103521 An introduction to his work at the time of his Fields medal award.
 Living people
 1944 births
 20thcentury mathematicians
 21stcentury mathematicians
 Algebraic geometers
 Belgian mathematicians
 Fields Medalists
 Abel Prize laureates
 Wolf Prize in Mathematics laureates
 Number theorists
 Viscounts of Belgium
 People from Brussels
 Free University of Brussels alumni
 Institute for Advanced Study faculty
 Members of the French Academy of Sciences
 Members of the Royal Swedish Academy of Sciences
 Members of the United States National Academy of Sciences
 Members of the Norwegian Academy of Science and Letters
 Foreign Members of the Russian Academy of Sciences