Math Labs

Pierre Deligne

fl. 1944, Etterbeek, Brussels, Belgium

AlgebraGeometryArithmetic and number theoryFields Medal 1978Crafoord Prize in Mathematics 1988Wolf Prize in Mathematics 2008Abel Prize 2013

Belgian mathematician and a principal architect of modern arithmetic geometry; proved the Weil conjectures and shaped étale cohomology, mixed Hodge theory and the theory of perverse sheaves.

Pierre René Deligne was born on 3 October 1944 in Etterbeek, Brussels, and discovered mathematics as a schoolboy when a teacher lent him a multi-volume algebra treatise. He studied at the Université libre de Bruxelles, where Jacques Tits became his mentor, and in 1965 moved to Paris to join Alexander Grothendieck's seminar at the IHÉS, also working closely with Jean-Pierre Serre. He received his doctorate from the Université Paris-Sud in 1972 and was a permanent member of the IHÉS from 1970 to 1984.

Deligne's central achievement is the proof of the Weil conjectures, completed in 1973 — estimates for the number of points of algebraic varieties over finite fields whose deepest clause is an analogue of the Riemann hypothesis for Frobenius eigenvalues. Beyond it he created mixed Hodge theory, developed the theory of weights and the l-adic Fourier transform in 'La conjecture de Weil II', proved the Ramanujan–Petersson conjecture for modular forms as a consequence, and co-founded the theory of perverse sheaves and the Riemann–Hilbert correspondence for connections with regular singularities.

In 1984 he moved to the Institute for Advanced Study in Princeton, where he is now professor emeritus, and the Belgian king created him a viscount in 2006. His influence extends through representation theory, the theory of motives and the Langlands program; the Abel Prize citation of 2013 described his contributions to algebraic geometry as having a transformative impact on number theory, representation theory and related fields.

Workplaces: Institut des Hautes Études Scientifiques (IHÉS), Institute for Advanced Study, Princeton

Belgium

Award-winning work

Fields Medal 1978For his solution of the Weil conjectures. Between 1968 and 1973 Deligne completed the proof of the conjectures posed by André Weil in 1949 on the number of points of algebraic varieties over finite fields; the hardest clause was the analogue of the Riemann hypothesis for the eigenvalues of the Frobenius endomorphism. Building on Grothendieck's étale cohomology, he combined monodromy arguments over Lefschetz pencils with estimates of Kloosterman-type exponential sums in 'La conjecture de Weil I' (1974), then settled the general case in 'La conjecture de Weil II' (1980). The result built a bridge between the arithmetic of finite fields and the topology of complex varieties that transformed modern algebraic geometry and number theory.

Contributions, linked to the library

Library topicAlgebraic geometryHis proof of the Weil conjectures and his work on étale cohomology and perverse sheaves reshaped algebraic geometry.Library topicArithmetic geometryThe Weil conjectures created a bridge between finite fields and complex geometry at the heart of arithmetic geometry.Library topicThe Riemann zeta function and the Riemann hypothesisThe deepest Weil conjecture is an analogue of the Riemann hypothesis for zeta functions of varieties over finite fields, which Deligne proved.