Laurent Lafforgue

fl. 1966, Antony, Hauts-de-Seine, France

Arithmetic and number theoryAlgebraAnalysisClay Research Award 2000Fields Medal 2002

French mathematician who proved the Langlands correspondence for the general linear groups over function fields, the deepest completed case of the Langlands program.

Laurent Lafforgue was born in Antony near Paris and entered the École Normale Supérieure in 1986. He completed his doctorate in 1994 at the Université Paris-Sud under Gérard Laumon, on Drinfeld shtukas — the geometric objects introduced by Vladimir Drinfeld to attack the Langlands correspondence over function fields. He became a CNRS researcher and then, in 2000, a permanent professor at the Institut des Hautes Études Scientifiques.

The Langlands program conjectures a deep correspondence between Galois representations and automorphic representations. Drinfeld had proved it for GL_2 over function fields in the 1970s, but the case of general linear groups of arbitrary rank required a new generation of ideas. Building on Drinfeld's shtukas and on the compactification and cohomology machinery developed with Laumon, Lafforgue established, in a series of works culminating around 1999–2000, the full global Langlands correspondence for GL_r over any global function field of positive characteristic. He received the Clay Research Award in 2000 and the Fields Medal in 2002 in Beijing.

After the Langlands work, Lafforgue turned to other directions, including noncommutative geometry, analytic invariants of groups (such as property (T) and the Baum–Connes conjecture) and, more recently, categorical foundations via topos theory. He has also been a prominent public voice in France on education and science policy, and was appointed to the Conseil d'État's advisory structures on scientific and ethical questions.

Workplaces: Institut des Hautes Études Scientifiques (IHÉS), Université Paris-Sud (Orsay)

France

Award-winning work

Fields Medal 2002Lafforgue received the medal for proving the global Langlands correspondence for the general linear groups GL_r over function fields of positive characteristic, completing the work begun by Drinfeld for GL_2. The Langlands correspondence predicts a bijection between n-dimensional Galois representations of a function field and cuspidal automorphic representations; Lafforgue established it for all r by studying the moduli stacks of Drinfeld shtukas and computing their cohomology. His proof combined the arithmetic geometry of these stacks, which encode the Galois side, with deep analytic techniques on the automorphic side, and it remains the only case where the full correspondence is known.

Contributions, linked to the library

Library topicModular formsAutomorphic forms on GL_r over function fields, the analytic side of his correspondence, are the function-field analogues of modular forms.

Library topicly-thuyet-so-dai-soThe Langlands correspondence links Galois representations of function fields to automorphic representations, the core theme of algebraic number theory.