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After completing a doctorate, he worked with Alexander Grothendieck at the Institut des Hautes Etudes Scientifiques near Paris, initially on the generalisation of Zariski's main theorem . He also 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 theoryA string theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. For this reason, string theories are able.
From 1970 until 1984, when he moved to the Institute for Advanced StudyThe Institute for Advanced Study is a private institution in Princeton Township, New Jersey, designed to foster pure cutting-edge research by scientists in a variety of fields without the complications of teaching or funding, or the agendas of sponsorship in Princeton, Deligne was a permanent member of the IHES staff. During this time he did much important work, besides the proof of the Weil conjectures: in particular with Lusztig on the use of etale cohomology to construct representations of algebraic groupIn algebraic geometry, two important classes of algebraic group arise, that for the most part are studied separately. The general definition of algebraic group is the expected one: a group in the category of algebraic varieties; or, more simply, a group ws, and with Rapoport on the moduli spaces from the 'fine' arithmetic point of view. He received a Fields MedalThe Fields Medal is a prize awarded to up to four mathematicians (not over forty years of age) at each International Congress of International Mathematical Union, since 1936 and regularly since 1948 at the initiative of the Canadian mathematican John Char in 1978.
In terms of the completion of some of the underlying Grothendieck programme 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 round the lack of knowledge of the Hodge conjectureHomology theory Algebraic geometry Conjectures The Hodge conjecture is a major unsolved problem of algebraic geometry. It is a conjectural description of the link between the algebraic topology of a non-singular complex algebraic variety, and its geometry, for some applications. He reworked the tannakian category theory in his paper for the Grothendieck Festschrift, employing Beck's theoremThere are two (completely different) theorems in mathematics (by two different mathematicians) going under the name of Beck's theorem . Beck's theorem in category theory Beck's monadicity theorem asserts that a functor : is monadic if and only if # U has – 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 theoryIn mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M''. It was developed by W. and the l-adic Galois representations. The Shimura variety theory is related, by the idea that such varieties should parametrise 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.
See also: Deligne conjecture, Deligne cohomology .
Deligne, Pierre Deligne, Pierre
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