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. Image File history File links Download high resolution version (1644x1245, 282 KB) Summary Pierre Deligne (born 3 October 1944), Belgian mathematician, at present member of the IAS, Princeton. ... Image File history File links Download high resolution version (1644x1245, 282 KB) Summary Pierre Deligne (born 3 October 1944), Belgian mathematician, at present member of the IAS, Princeton. ... October 3 is the 276th day of the year (277th in Leap years). ... 1944 (MCMXLIV) was a leap year starting on Saturday (link will take you to calendar). ... This article is in need of attention from an expert on the subject. ... In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... 1973 (MCMLXXIII) was a common year starting on Monday. ... Bold textItalic textBold text // Headline text Emblem of the Brussels-Capital Region Flag of The City of Brussels Brussels (French: Bruxelles, Dutch: Brussel, German: Brüssel) is the capital of Belgium, the French community of Belgium, the Flemish community and of the European Union. ...

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. Alexander Grothendieck (Berlin, March 28, 1928) was one of the most important mathematicians active in the 20th century. ... IHÉS main building The Institut des Hautes Études Scientifiques (I.H.É.S.) is a French institute supporting advanced research in mathematics and theoretical physics. ... The Eiffel Tower has become a symbol of Paris throughout the world. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ... In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles...

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. Fuld Hall The Institute for Advanced Study is a private institution in Princeton Township, New Jersey, U.S.A. (although it is not part of Princeton University), designed to foster pure cutting-edge research by scientists in a variety of fields without the complications of teaching or funding, or the... George Lusztig is an American mathematician. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... Modular form - Wikipedia /**/ @import /skins-1. ... The 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 (therefore once every four years), since 1936 and regularly since 1950 at the initiative of the Canadian mathematician John Charles Fields. ...

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. In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... The Hodge conjecture is a major unsolved problem of algebraic geometry. ... In category theory, a branch of mathematics, Becks monadicity theorem asserts that a functor is monadic if and only if U has a left adjoint; U reflects isomorphisms; and C has co-equalizers of U-split co-equalizer pairs, and U preserves those co-equalizers. ... In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric... In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ...

Deligne has written a book with G. D. Mostow on monodromy. He was awarded the Crafoord Prize in 1988. In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ... The Crafoord Prize was established in 1980 by Holger Crafoord, the inventor of the artificial kidney, and his wife Anna-Greta Crafoord. ...

See also

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

In mathematics, there are a number of so-called Deligne conjectures, provided by Pierre Deligne. ... In mathematics, the Deligne-Mumford moduli space of curves is a refined construction of a moduli space of algebraic curves, that is work from 1969 by Pierre Deligne and David Mumford. ...