The study and classification of Lagrangian submanifolds in symplectic manifolds are central topics of modern symplectic topology. In particular, we focus on the Lagrangian knot problem, which asks whether two given Lagrangians are isotopic. We explore how symplectic topology has developed together with understanding of Lagrangian submanifolds.
In the first part, we study the knottedness of Lagrangian tori, discovering exotic Lagrangian tori such as the Chekanov and Vianna tori.
In the second part, we revisit a well-known result of the unknottedness of Lagrangian spheres in del Pezzo surfaces, as established by Evans.