Jim Carlson was President of the Clay Mathematics Institute 2003-2012.
The 2008 Clay Research Award was made to Clifford Taubes for his proof of the Weinstein conjecture in dimension three.
The Weinstein conjecture is a conjecture about the existence of closed orbits for the Reeb vector field on a contact manifold. A contact manifold is an odd-dimensional manifold with a one-form A such that A wedged with the n-th exterior power of dA is everywhere nonzero. In particular, the kernel of A is a maximally nonintegrable field of hyperplanes in the tangent bundle. The Reeb vector field generates the kernel of dA and pairs to one with A. Alan Weinstein asked some thirty years ago whether this vector field must, in all cases, have a closed orbit. (The unit sphere in complex n-space with A the annihilator of the maximal complex subspace of the real tangent space is an example of a contact manifold and contact 1-form. In this case, the orbits of the Reeb vector field generate the circle action whose quotient gives the associated complex projective space.) Note, by contrast, that there exist non-contact vector fields, even on the 3-sphere, with no closed orbits. These are the counter-examples (due to Schweitzer, Harrison and Kuperberg) to the Seifert conjecture. Hofer affirmed the Weinstein conjecture in many 3-dimensional cases, for example the three-sphere and contact structures on any 3 dimensional, reducible manifold. Taubes’ affirmative solution of the Weinstein conjecture for any 3-dimensional contact manifold is based on a novel application of the Seiberg-Witten equations to the problem.
The 2008 Clay Research Award was made to Claire Voisin for her disproof of the Kodaira conjecture.
The Kodaira conjecture was formulated in 1960, when Kunihiko Kodaira showed that any compact complex Kaehler surface can be deformed to a projective algebraic surface. For the proof, Kodaira used his classification theorem for complex surfaces. The conjecture asks whether Kaehler manifolds of higher dimension can be deformed to a projective algebraic manifold. Voisin constructs counterexamples: in each dimension four or greater, there is a compact Kaehler manifold which is not homotopy equivalent to a projective one. For dimension at least six, she gives examples which are also simply connected. A later result gives a substantial strengthening: in any even dimension ten or greater, there exist compact Kaehler manifolds, no bimeromorphic model of which is homotopy equivalent to a projective algebraic variety. Distinguishing the homotopy type of projective and non-projective Kaehler manifolds is achieved through novel Hodge-theoretic arguments that place subtle restrictions on the topological intersection ring of a projective manifold.
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 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 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.
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.