Danny Calegari

The 2009 Clay Research Award was made to Ian Agol, Danny Calegari and David Gabai for their solutions of the Marden tameness conjecture, and, by implication through the work of Thurstan and Canary, of the Ahlfors measure conjecture.

The Langlands program is a collection of conjectures and theorems that unify the theory of automorphic forms, relating it intimately to the main stream of number theory, with close relations to harmonic analysis on algebraic groups as well as arithmetic algebraic geometry. Since its origins in the winter of 1966-67, when it was laid out in a letter from Langlands to André Weil, it has served as the basis of much deep work, including applications to many famous problems in number theory, e.g., Artin’s conjectures on L-functions, Fermat’s Last Theorem, and the behaviour of Hasse-Weil zeta functions.

The tameness conjecture asserts that a hyperbolic 3-manifold with finitely-generated fundamental group is homeomorphic to the interior of a compact 3-manifold (possibly with boundary). The Ahlfors conjecture asserts that the limit set of a finitely generated Kleinian group (i.e. the minimal invariant set on the Riemann sphere, which is the boundary at infinity of hyperbolic 3-space) has either full or zero measure, and in the former case the action of the group on it is ergodic.