Category: Research Award Winners
Affiliation: Harvard University
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.