Fillability of contact surgeries and Lagrangian discs

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties of a contact structure are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact (r) surgery is symplectically/Stein fillable for r in (0,1]. This is joint work with James Conway and John Etnyre.