UCLA

In this thesis we prove some classification results for symplectic and exact Lagrangian fillings in contact geometry. First we prove a classification result for symplectic fillings of certain contact manifolds. Let $(M,\xi)$ be a contact 3-manifold and $T^2 \subset (M,\xi)$ a mixed torus. We prove a JSJ-type decomposition theorem for strong and exact symplectic fillings of $(M,\xi)$ when $(M,\xi)$ is cut along $T^2$. As an application we prove the uniqueness of exact fillings when $(M,\xi)$ is obtained by Legendrian surgery on a knot in $(S^3,\xi_{std})$ which is stabilized both positively and negatively. Second we show a classification result for Lagrangian fillings of Legendrian representatives of positive braid closures in $S^3$. This second result follows from an injectivity result for augmentation categories of positive braids.