Building Lagrangian fillings

In the marriage of low dimensional topology and symplectic topology, there lives Legendrians knots and their exact Lagrangian fillings. Legendrian knots are smooth knots satisfying some tangency conditions, such that their projection diagrams have certain cusps. The Lagrangian fillings are filling surfaces of knots in the 4-dimensional ball, again satisfying some extra conditions. Legendrian knots and Lagrangian fillings are useful objects to understand 3-dimensional contact topology and 4-dimensional symplectic topology. In this talk, I will discuss some work with Roger Casals on the construction of exact Lagrangian fillings.