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ⓘ Pierre Deligne




Pierre Deligne
                                     

ⓘ Pierre Deligne

Pierre Rene, Viscount Deligne is a Belgian mathematician. He is best 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.

                                     

1. Early life and education

Deligne was born in Etterbeek, attended school at Athenee Adolphe Max and studied at the Universite libre de Bruxelles ULB, writing a dissertation titled Theoreme de Lefschetz et criteres de degenerescence 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 Theorie de Hodge.

                                     

2. Career

Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Etudes Scientifiques IHES near Paris, initially on the generalization within scheme theory of Zariskis 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. Delignes 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 Delignes 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. Delignes 1974 paper contains the first proof of the Weil conjectures, Delignes contribution being to supply the estimate of the eigenvalues of the Frobenius endomorphism, considered the geometric analogue of the Riemann hypothesis. Delignes 1980 paper contains a much more general version of the Riemann hypothesis.

From 1970 until 1984, Deligne was a permanent member of the IHES staff. During this time he did much important work outside of his work on algebraic geometry. In joint work with George Lusztig, Deligne applied etale 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.

                                     

2.1. Career Hodge cycles

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. The theory of mixed Hodge structures being a powerful construction around the conjecture. He reworked the Tannakian category theory in his 1990 paper for the Grothendieck Festschrift, employing Becks 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.



                                     

2.2. Career Perverse sheaves

With Alexander Beilinson, Joseph Bernstein, and Ofer Gabber, Deligne made definitive contributions to the theory of perverse sheaves. This theory plays an important role in the recent proof of the fundamental lemma by Ngo Bảo Chau. It was also used by Deligne himself to greatly clarify the nature of the Riemann-Hilbert correspondence, which extends Hilberts twenty-first problem to higher dimensions.

                                     

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, "for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields". He was elected a foreign member of the Academie des Sciences de Paris in 1978.

In 2006 he was ennobled by the Belgian king as viscount.

In 2009, Deligne was elected a foreign member of the Royal Swedish Academy of Sciences. He is a member of the Norwegian Academy of Science and Letters.

                                     

4. Selected publications

  • Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis 1975. "Real homotopy theory of Kahler manifolds". Inventiones Mathematicae. 29 3: 245–274. Bibcode:1975InMat.29.245D. doi:10.1007/BF01389853. MR 0382702.
  • Deligne, Pierre 1990. "Categories tannakiennes". Grothendieck Festschrift Vol II. Progress in Mathematics. 87: 111–195.
  • Deligne, Pierre; Mostow, George Daniel 1993. Commensurabilities among Lattices in PU1,n. Princeton, N.J.: Princeton University Press. ISBN 0-691-00096-4.
  • Deligne, Pierre 1974. "La conjecture de Weil: I". Publications Mathematiques de lIHES. 43: 273–307. doi:10.1007/bf02684373.
  • 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.
  • Deligne, Pierre 1980. "La conjecture de Weil: II". Publications Mathematiques de lIHES. 52: 137–252. doi:10.1007/BF02684780.


                                     

5. Hand-written letters

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

  • "Delignes letter to Piatetskii-Shapiro 1973" PDF. Archived from the original PDF on 7 December 2012. Retrieved 15 December 2012.
  • "Delignes letter to Looijenga 1974" PDF. Retrieved 20 January 2020.
  • "Delignes letter to Jean-Pierre Serre around 1974". 15 December 2012.
                                     

6. Concepts named after Deligne

The following mathematical concepts are named after Deligne:

  • Deligne motive
  • Fourier–Deligne transform
  • Deligne cohomology
  • Deligne–Lusztig theory
  • Deligne tensor product of abelian categories denoted ⊠ {\displaystyle \boxtimes }
  • Langlands–Deligne local constant
  • Deligne–Mumford stacks
  • Deligne–Mumford moduli space of curves

Additionally, many different conjectures in mathematics have been called the Deligne conjecture:

  • 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 monodromy, also known as the weight monodromy conjecture, or purity conjecture for the monodromy filtration.
  • The Deligne conjecture in deformation theory is about the operadic structure on Hochschild cohomology. It was proved by Maxim Kontsevich and Yan Soibelman, James E. McClure and Jeffrey H. Smith and others. It is of importance in relation with string theory.
  • There is a Deligne conjecture on 1-motives arising in the theory of motives in algebraic geometry.
  • There is a Deligne–Langlands conjecture of historical importance in relation with the development of the Langlands philosophy.
  • There is a Deligne conjecture in the representation theory of exceptional Lie groups.
  • Delignes conjecture on the Lefschetz trace formula now called Fujiwaras theorem for equivariant correspondences.
  • There is a Gross–Deligne conjecture in the theory of complex multiplication.
                                     
  • his Ph.D. in 1973 under the supervision of Peter Swinnerton - Dyer and Pierre Deligne Reid was a research fellow of Christ s College, Cambridge from 1973
  • where he is now a Professor. He also worked with David Kazhdan and Pierre Deligne Born 1955 in Kfar - Saba, raised in Ramat - Gan, Flicker studied Mathematics
  • Ph.D. from Paris - Sud 11 University in 1976, under the supervision of Pierre Deligne He currently holds a chair for arithmetic algebraic geometry at the
  • Hitchin. The Deligne Simpson Problem, an algebraic problem associated with monodromy matrices, is named after Carlos Simpson and Pierre Deligne Simpson was
  • the Weil Conjecture, Berlin: Springer - Verlag, ISBN 978 - 0 - 387 - 12175 - 8 Deligne Pierre Katz, Nicholas, eds. 1973 Seminaire de Geometrie Algebrique du
  • provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne From 1959 onward Serre s interests turned towards group theory, number
  • Tits 1966: No award 1970: J.A. Thas 1974: Pierre Deligne 1978: Michel Cahen 1982: Francis Buekenhout 1986: Pierre Lecomte 1990: Luc Haine 1994: Luc Lemaire
  • three - dimensional Calabi - Yau varieties mixed Hodge structures after Pierre Deligne and variation of Hodge structures after Phillip Griffiths Steenbrink
  • in 1997 in favor of the Grande Medaille. 1954 Georges Valiron 1974 Pierre Deligne 1992 John G. Thompson List of things named after Henri Poincare List

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