Abstract: The Ozsváth-Szabó contact invariant is a helpful tool to detect tightness of contact manifolds. To any contact structure, it associates an element in the Heegaard Floer cohomology of the underlying manifold. I'll explain how to compute the invariant for positive surgeries along Legendrian knots in the standard 3-sphere. In particular, necessary and sufficient condition for its nonvanishing will be given.

The first part of the talk will be an introduction to some aspects of contact topology and Heegaard Floer homology. After giving the result of the computation mentioned above, I'll discuss a Legendrian cabling construction.

The proof will be sketched in the second part, after introducing some tools in Floer theory (Sutured Floer homology, relative contact invariants, gluing maps).