Lagrangian submanifold rigidity has been a fundamental topic in symplectic topology, contributing to key theories like the Arnold-Givental conjecture and Lagrangian Floer theory. These theories often show that intersections between Lagrangian submanifolds are unavoidable via symplectic maps, exemplified by Biran's concept of Lagrangian Barriers (2001).
Conversely, submanifolds not containing Lagrangian submanifolds usually exhibit flexibility, and can often be symplectically displaced. In this joint work with Richard Hind and Yaron Ostrover, we introduce what appears to be the first illustration of Symplectic Barriers, demonstrating necessary intersections of symplectic embeddings with symplectic (non-Lagrangian) submanifolds. The key point is that Lagrangian submanifolds are not the sole barriers, and there exist situations where a symplectic submanifold is not flexible.

In our work, we also answer a question by Sackel–Song–Varolgunes–Zhu and calculate the optimal symplectic ball embedding in the ball after removing a codimension 2 hyperplane with a prescribed Kähler angle.