Pierre René, Viscount Deligne (born 3 October 1944 in
BrusselsBrussels , officially the Brussels Region or Brussels-Capital Region , is the de facto capital city of the European Union and the largest urban area in Belgium...
) is a
BelgianThe Kingdom of Belgium is a country in northwest Europe. It is a founding member of the European Union and hosts its headquarters, as well as those of other major international organizations, including NATO...
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
. He is known for fundamental work on the
Weil conjecturesIn mathematics, the Weil conjectures, which had become theorems by 1974, were some highly-influential proposals from the late 1940s by André Weil on the generating functions derived from counting the number of points on algebraic varieties over finite fields.A variety V over a finite field with q...
, leading finally to a complete proof in 1973.
Life
He was born in
BrusselsBrussels , officially the Brussels Region or Brussels-Capital Region , is the de facto capital city of the European Union and the largest urban area in Belgium...
, and studied at the
Universite Libre de BruxellesThe Université Libre de Bruxelles is a French-speaking university in Brussels, Belgium. It has about 20,000 students. It was ranked 54th worldwide in the THES 2004 but only 183rd in 2008...
(ULB).
After completing a
doctorateA doctorate is an academic degree or professional degree that in most countries represents the highest level of formal study or research in a given field. In some countries it also refers to a class of degrees which qualify the holder to practice in a specific profession . The best-known example...
under the supervision of
Alexander GrothendieckAlexander Grothendieck is considered one of the greatest mathematicians of the 20th century.He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory,...
, he worked with him at the
Institut des Hautes Études ScientifiquesThe Institut des Hautes Études Scientifiques is a French institute supporting advanced research in mathematics and theoretical physics...
(IHÉS) near
ParisParis is the capital of France and the country's most populous city. It is situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, initially on the generalization within scheme theory of
Zariski's main theoremIn algebraic geometry, a field in mathematics, Zariski's main theorem, or Zariski's connectedness theorem, is a theorem proved by which implies that fibers over normal points of birational projective morphisms of varieties are connected. The theorem can be stated in several ways which at first...
. In 1968, he also worked with
Jean-Pierre SerreJean-Pierre Serre is a French mathematician in the fields of algebraic geometry, number theory and topology. He has received numerous awards and honors for his mathematical research and exposition, including the Fields Medal in 1954 and the Abel Prize in 2003.-Early years:Born in Bages,...
; their work led to important results on the l-adic representations attached to
modular formIn mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s, and the conjectural
functional equationIn mathematics or its applications, a functional equation is any equation that specifies a function in implicit form .Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types...
s of
L-functionThe theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s. Deligne's also focused on topics in
Hodge TheoryIn mathematics, Hodge theory, named after W. V. D. Hodge, 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...
. He introduced
weightsWeights are exercise equipment used for strength training. The term is typically used as a shortened form of the term free weights, but it can also refer to any exercise machine that uses weighted plates to generate the major opposing force....
and tested them on objects in
complex geometryIn mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
. He also collaborated with
David MumfordDavid Bryant Mumford is a mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory...
on a new description of the
moduli spaceIn algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s for curves. Their work came be seen as a brilliant introduction to algebraic stacks, and recently has been applied to questions arising from
string theoryString theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...
. Perhaps Deligne's most famous contribution was his proof of the third and last of the
Weil ConjecturesIn mathematics, the Weil conjectures, which had become theorems by 1974, were some highly-influential proposals from the late 1940s by André Weil on the generating functions derived from counting the number of points on algebraic varieties over finite fields.A variety V over a finite field with q...
. As a corollary he proved that the celebrated Ramanujan-Peterrson conjecture on modular forms.
From 1970 until 1984, when he moved to the
Institute for Advanced StudyThe Institute for Advanced Study, located in Princeton, New Jersey, United States, is a center for theoretical research and intellectual inquiry. The Institute is perhaps best known as the academic home of Albert Einstein, John von Neumann, and Kurt Gödel, after their immigration to the United...
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 LusztigGeorge Lusztig is a Romanian-born American mathematician. He is a Norbert Wiener Professor at the Department of Mathematics, MIT.Born in Timişoara, he did his undergraduate studies at the University of Bucharest...
, Deligne and Lusztig applied
étale cohomologyIn mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
to construct representations of
finite groups of Lie typeIn mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
; with Rapoport, Deligne worked on the moduli spaces from the 'fine' arithmetic point of view, with application to
modular formIn mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s. He received a
Fields MedalThe Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. The Fields Medal is often viewed as the top honor a mathematician can receive. It...
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 conjectureThe Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
, for some applications. He reworked the
tannakian categoryIn mathematics, a tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K...
theory in his paper for the
Grothendieck Festschrift, employing
Beck's theoremIn category theory, a branch of mathematics, Beck's monadicity theorem asserts that a functoris monadic if and only if# U has a left adjoint;# U reflects isomorphisms; and...
– 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, named after W. V. D. Hodge, 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...
and the l-adic Galois representations. The
Shimura varietyIn mathematics, a Shimura variety is an analogue of a modular curve, and is a quotient of an Hermitian symmetric space by a congruence subgroup of an algebraic group. The simplest example is the quotient of the upper half-plane by SL2...
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-theoryIn mathematics, K-theory is a tool used in several disciplines. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It also has some applications in operator algebras...
approaches.
He was awarded the
Fields MedalThe Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. The Fields Medal is often viewed as the top honor a mathematician can receive. It...
in 1978, the
Crafoord PrizeThe Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord...
in 1988, the
Balzan PrizeThe International Balzan Prize Foundation awards four annual monetary prizes to people or organisations who have made outstanding achievements in the fields of humanities, natural sciences, culture, as well as for endeavours for peace and the brotherhood of man....
in 2004 and the
Wolf PrizeThe Wolf Prize is an international award, that has been presented annually since 1978 to living scientists and artists for "achievements in the interest of mankind and friendly relations among peoples ... irrespective of nationality, race, colour, religion, sex or political views."The prize is...
in 2008.
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 SciencesThe Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademin is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2 June...
.
See also
- Deligne conjecture
In mathematics, there are a number of so-called Deligne conjectures, provided by Pierre Deligne. These are independent conjectures in various fields of mathematics....
- Deligne-Mumford moduli space of curves
In algebraic geometry, a moduli space of curves is a geometric space whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space...
- Deligne cohomology
External links