Mathematical Conversations

Given a smooth knot in the 3-sphere, there are many topologically distinct smooth surfaces in the 4-ball that have this knot as its boundary. However, if the knot is Legendrian, meaning that it satisfies a geometrical condition imposed by a contact structure, and the surface is Lagrangian, meaning that it satisfies a geometrical condition imposed by a symplectic structure, then the Legendrian knot determines the topology of any of its Lagrangian fillings. I will give a brief introduction to Legendrian knots, Lagrangian fillings, and the Seidel Isomorphism, which says that, in arbitrary dimensions, the Legendrian determines the homology of any Lagrangian filling.