2008 Clay Research Awards Announced
February 21, 2008
The Clay Mathematics Institute announces the 2008 Clay Research Awards.
Cliff Taubes, Harvard University
For his proof of the Weinstein conjecture in dimension three.
Claire Voisin, CNRS, IHES, and the Institut Mathématique de Jussieu
For her disproof of the Kodaira conjecture.
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 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.
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