Pierre Deligne | |
|---|---|
_(cropped).jpg) Deligne in March 2005 | |
| Born | (1944-10-03)3 October 1944 (age 81) Etterbeek, Belgium |
| Education | Université libre de Bruxelles (BS) Paris-Sud University (MS,PhD) |
| Known for | Proof of theWeil conjectures Perverse sheaves Concepts named after Deligne |
| Awards | Abel Prize (2013) Wolf Prize (2008) Balzan Prize (2004) Crafoord Prize (1988) Fields Medal (1978) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Institute for Advanced Study Institut des Hautes Études Scientifiques |
| Doctoral advisor | Alexander Grothendieck |
| Doctoral students | Lê Dũng Tráng Miles Reid Michael Rapoport |
Pierre René, Viscount Deligne (French:[dəliɲ]; born 3 October 1944) is a Belgian mathematician. He is best known for work on theWeil conjectures, leading to a complete proof in 1973. He is the winner of the 1978Fields Medal, 1988Crafoord Prize, 2008Wolf Prize and 2013Abel Prize.
Deligne was born inEtterbeek, attended school atAthénée Adolphe Max and studied at theUniversité libre de Bruxelles (ULB), writing a dissertation titledThéorème de Lefschetz et critères de dégénérescence de suites spectrales (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at theUniversity of Paris-Sud inOrsay 1972 under the supervision ofAlexander Grothendieck, with a thesis titledThéorie de Hodge.
Starting in 1965, Deligne worked with Grothendieck at theInstitut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization withinscheme theory ofZariski's main theorem. In 1968, he also worked withJean-Pierre Serre; their work led to important results on the l-adic representations attached tomodular forms, and the conjecturalfunctional equations ofL-functions. Deligne also focused on topics inHodge theory. He introduced the concept of weights and tested them on objects incomplex geometry. He also collaborated withDavid Mumford on a new description of themoduli spaces for curves. Their work came to be seen as an introduction to one form of the theory ofalgebraic stacks, and recently has been applied to questions arising fromstring theory.[1] But Deligne's most famous contribution was his proof of the third and last of theWeil conjectures. This proof completed a programme initiated and largely developed byAlexander Grothendieck lasting for more than a decade. As a corollary he proved the celebratedRamanujan–Petersson conjecture formodular forms of weight greater than one; weight one was proved in his work with Serre. Deligne's 1974 paper contains the first proof of theWeil conjectures. Deligne's contribution was to supply the estimate of theeigenvalues of theFrobenius endomorphism, considered the geometric analogue of theRiemann hypothesis. It also led to a proof of theLefschetz hyperplane theorem and the old and new estimates of the classical exponential sums, among other applications. 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 withGeorge Lusztig, Deligne appliedétale cohomology to construct representations offinite groups of Lie type; withMichael Rapoport, Deligne worked on the moduli spaces from the 'fine' arithmetic point of view, with application tomodular forms. He received aFields Medal in 1978. In 1984, Deligne moved to theInstitute for Advanced Study in Princeton.
In terms of the completion of some of the underlying Grothendieck program of research, he definedabsolute Hodge cycles, as a surrogate for the missing and still largely conjectural theory ofmotives. This idea allows one to get around the lack of knowledge of theHodge conjecture, for some applications. The theory ofmixed Hodge structures, a powerful tool in algebraic geometry that generalizes classical Hodge theory, was created by applying weight filtration, Hironaka'sresolution of singularities and other methods, which he then used to prove the Weil conjectures. He reworked theTannakian category theory in his 1990 paper for the "Grothendieck Festschrift", employingBeck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimateWeil cohomology. All this is part of theyoga of weights, unitingHodge theory and the l-adicGalois representations. TheShimura 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 usedK-theory approaches.
WithAlexander Beilinson,Joseph Bernstein, andOfer Gabber, Deligne made definitive contributions to the theory ofperverse sheaves.[2] This theory plays an important role in the recent proof of thefundamental lemma byNgô Bảo Châu. It was also used by Deligne himself to greatly clarify the nature of theRiemann–Hilbert correspondence, which extendsHilbert's twenty-first problem to higher dimensions. Prior to Deligne's paper,Zoghman Mebkhout's 1980 thesis and the work ofMasaki Kashiwara throughD-modules theory (but published in the 80s) on the problem have appeared.
In 1974 at the IHÉS, Deligne's joint paper withPhillip Griffiths,John Morgan andDennis Sullivan on the realhomotopy theory of compactKähler manifolds was a major piece of work in complex differential geometry which settled several important questions of both classical and modern significance. The input from Weil conjectures, Hodge theory, variations of Hodge structures, and many geometric and topological tools were critical to its investigations. His work in complexsingularity theory generalizedMilnor maps into an algebraic setting and extended thePicard-Lefschetz formula beyond their general format, generating a new method of research in this subject. His paper withKen Ribet on abelian L-functions and their extensions toHilbert modular surfaces and p-adic L-functions form an important part of his work inarithmetic geometry. Other important research achievements of Deligne include the notion of cohomological descent, motivic L-functions, mixed sheaves, nearbyvanishing cycles, central extensions ofreductive groups, geometry and topology ofbraid groups, providing the modern axiomatic definition of Shimura varieties, the work in collaboration withGeorge Mostow on the examples of non-arithmetic lattices and monodromy ofhypergeometric differential equations in two- and three-dimensional complexhyperbolic spaces, etc.
He was awarded theFields Medal in 1978, theCrafoord Prize in 1988, theBalzan Prize in 2004, theWolf Prize in 2008, and theAbel 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 asviscount.[3]
In 2009, Deligne was elected a foreign member of theRoyal Swedish Academy of Sciences[4] and a residential member of theAmerican Philosophical Society.[5] He is a member of theNorwegian Academy of Science and Letters.[6]
Deligne wrote multiple hand-written letters to other mathematicians in the 1970s. These include
The following mathematical concepts are named after Deligne:
Additionally, many different conjectures in mathematics have been called theDeligne conjecture:
