Title: Slice links and smooth 4-manifolds Abstract: We vary the trace embedding lemma in order to prove results about smooth, closed, simply connected 4-manifolds, studying smoothly slice links in them. We focus on homotopy 4-spheres, which are potential counterexamples to the smooth 4-dimensional Poincaré conjecture. In particular, we split them, as for exotic R^4's,  in large and

Title: Infinitely many Lagrangian Tori in Milnor fibers constructed via Lagrangian Fillings of Legendrian links Abstract: One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings

Title: The homotopy type of a finite 2-complex with non-minimal Euler characteristic Abstract: Two presentations for a group G which have the same deficiency are called exotic if the corresponding presentation complexes are not homotopy equivalent. The first examples of exotic presentations were found by Dunwoody and Metzler in the 1970s but, owing to the difficulty of