1. Introduction

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.

2. Biography

Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled Théorème de Lefschetz et critères de dégénérescence de suites spectrales. He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled Théorie de Hodge.

Starting in 1972, Deligne worked with Grothendieck 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 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 the Frobenius endomorphism, 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, 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 1984, Deligne moved to the Institute for Advanced Study in Princeton.

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 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 is not yet a finished product, and more recent trends have used K-theory approaches.

3. 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]

4. Selected Publications

• Deligne, Pierre (1974). "La conjecture de Weil: I". Publications Mathématiques de l'IHÉS 43: 273–307. doi:10.1007/bf02684373. 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. doi:10.1007/BF02684780. http://www.numdam.org/item?id=PMIHES_1980__52__137_0. 
• Deligne, Pierre (1990). "Catégories tannakiennes". Grothendieck Festschrift vol II. Progress in Mathematics 87: 111–195. https://publications.ias.edu/deligne/paper/406. 
• Deligne, Pierre; Mostow, G. Daniel (1993). Commensurabilities among Lattices in PU(1,n). Princeton, N.J.: Princeton University Press. ISBN 0-691-00096-4. 
• 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:0-8218-1198-3.

5. Hand-Written Letters

Deligne wrote multiple hand-written letters to other mathematicians in the 1970s. These include

• "Deligne’s letter to Piatetskii-Shapiro (1973)". Archived from the original on 7 December 2012. https://web.archive.org/web/20121207001945/http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf. Retrieved 15 December 2012. 
• "Deligne’s letter to Jean-Pierre Serre (around 1974)". 2012-12-15. http://mathoverflow.net/questions/81739/delignes-letter-to-jean-pierre-serre. 
• "Deligne’s letter to Looijenga (1974)" (PDF). http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf. Retrieved 15 December 2012. 

6. 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 [math]\displaystyle{ \boxtimes }[/math])^[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 L-functions is a formulation of the hope for algebraicity of L(n) where L is an L-function and n is an integer in some set depending on L.
• There is a Deligne conjecture on 1-motives 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]