Viewing archives for 2009

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 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.

David Gabai

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 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.

Jean-Loup Waldspurger

The 2009 Clay Research Award was made to Jean-Loup Waldspurger for his work in p-adic harmonic analysis, particularly his contributions to the transfer conjecture and the fundamental lemma.  This work, combined with that of others, makes it possible to finally resolve important, long-standing parts of the Langlands program.

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.

Ian Agol

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 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.

Xinyi Yuan

Xinyi Yuan received his PhD from Columbia University in 2008 under the supervision of Shou-Wu Zhang.  In his 2006 preprint, “Big Line Bundles over Arithmetic Varieties,” he proves an arithmetic analogue of a theorem of Siu and derives, among other consequences, a natural sufficient condition for when the orbit under the absolute Galois group is equidistributed.  Xinyi was apponted as a Clay Research Fellow for a term of three years beginning July 2008.

Teruyoshi Yoshida

Teruyoshi Yoshida received his PhD in 2006 from Harvard University under the supervision of Richard Taylor. His mathematical interest is in the interface between automorphic forms and arithmetic algebraic geometry, with much of his work concerned with the geometric structure of Shimura varieties at places of bad reduction. Teruyoshi was appointed as a Clay Research Fellow for a term of three years beginning December 2007.

David Speyer

David Speyer received his PhD from the University of California, Berkeley in 2005 under the supervision of Bernd Sturmfels.  Much of his research is in the emerging area of tropical geometry, to which he has contributed both fundamental results as well as applications, e.g., a new proof of Horn’s conjecture on eigenvalues of hermitian matrices and (with Lior Pachter) the reconstruction of phylogentic trees from subtree weights. His research interests include continuing work in tropical geometry, cluster algebras and the geometry of grassmannians and flag varieties.  David was appointed as a Clay Research Fellow for a term of five years beginning June 2005.

Sucharit Sarkar

Sucharit Sarkar received his PhD from Princeton University in 2009 under the guidance of Zoltan Szabo.  His research area is in low dimensional topology. His dissertation addressed topics in Heegaard Floer homology for 3-manifolds and knots inside 3-manifolds.  Sucharit was appointed as a Clay Research Fellow for a term of five years beginning July 2009.

Samuel Payne

Samuel Payne received his PhD from the University of Michigan in 2006 under the supervision of William Fulton. His thesis,Toric vector bundles, gives a surprising and simple construction of complete toric varieties on which there are no nontrivial equivariant bundles of rank less than or equal to three.  Sam was appointed as a Clay Research Fellow for a term of four years beginning June 2006.

Sophie Morel

Sophie Morel received her PhD from Université Paris-Sud in 2005 under the supervision of Gérard Laumon.  Her thesis, Complexes d’intersection des compactifications de Baily-Borel – le cas des groupes unitaires sur Q is an important step forward in the Langlands program.  She develops a theory of weight truncation on varieties over finite fields with which she derives a simple description of the intersection complexes on the Baily-Borel compactifications of certain Shimura varieties over finite fields.  From this in turn she obtains a formula for the trace of the Frobenius endomorphism on the Euler characteristic of the intersection cohomology.  Sophie was appointed as a Clay Research Fellow for a term of five years beginning October 2006.

Davesh Maulik

Davesh Maulik received his PhD in 2007  from Princeton University under the supervision of Rahul Pandharipande.  His mathematical interests include algebraic geometry and its connections with symplectic geometry, mathematical physics, and combinatorics.  His research focus is in the area of Gromov-Witten theory and enumerative geometry.  Davesh was appointed as a Clay Research Fellow for a term of five years beginning July 2007.

Adrian Ioana

Adrian Ioana was born in Romania (1981), he received his B.S. from Unversity of Bucharest and his PhD in 2007 from UCLA under the direction of Professor Sorin Popa. His research interests are von Neumann algebras and orbit equivalence. Adrian, together with  Jesse Peterson and Sorin Popa, have proved the existence of a II_1 factor with trivial outer automorphism group.  He also showed that any countable group which contains a free subgroup has uncountably many non orbit equivalent actions.  Adrian was appointed as a Clay Research Fellow for a term of three years beginning August 2008.