The topology and geometry of contact structures in dimension three

The goal of this article is to survey recent developments in the theory of contact structures in dimension three.Comment: 13 pages. To appear in the ICM 2006 proceeding

Semi-global Kuranishi charts and the definition of contact homology

We define the contact homology algebra for any contact manifold and show that it is an invariant of the contact manifold. More precisely, given a contact manifold $(M,\xi)$ and some auxiliary data $\mathcal{D}$, we define an algebra $HC(\mathcal{D})$. If $\mathcal{D}_1$ and $\mathcal{D}_2$ are two choices of auxiliary data for $(M,\xi)$, then $HC(\mathcal{D}_1)$ and $HC(\mathcal{D}_2)$ are isomorphic. We use a simplified version of Kuranishi perturbation theory, consisting of semi-global Kuranishi charts.Comment: 95 pages. Proposition 9.4.2 was removed, as its proof has a gap. The main theorem is not affecte

Knots and Contact Geometry

We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy