# The many forms of rigidity for symplectic embeddings

# The many forms of rigidity for symplectic embeddings

We look at the following chain of symplectic embedding problems in dimension four.$E(1,a) \to Z^4(A) E(1,a) \to C^4(A) E(1,a) \to P(A,bA) (b \in \mathbb{N}_{\geq 2}) E(1,a) \to T^4(A)$

Here $E(1,a)$ is a symplectic ellipsoid, $Z^4(A)$ is the symplectic cylinder $D^2(A) \times \mathbb{R}^2, C^4(A) = D^2(A) \times D^2(A)$ is the cube and $P(A,bA) = D^2(A) \times D^2(bA)$ the polydisc, and $T^4(A) = T^2(A) \times T^2(A)$, where $T^2(A)$ is the 2-torus of area $A$. In each problem we ask for the smallest $A$ for which $E(1,a)$ symplectically embeds. The answer is very different in each case: total rigidity, total flexibility with a hidden rigidity, and a two-fold subtle transition between them. The talk is based on works by Cristofaro-Gardiner, Frenkel, Latschev, McDuff, Muller, and myself.