# The many forms of rigidity for symplectic embeddings

 Princeton/IAS Symplectic Geometry Seminar Topic: The many forms of rigidity for symplectic embeddings Speaker: Felix Schlenk Affiliation: University of Neuchâtel Date: Thursday, March 30 Time/Room: 9:30am - 10:30am/S-101 Video Link: https://video.ias.edu/puias/2017/0330-FelixSchlenk

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 $D_2(A) \times R_2$, $C_4(A) = D_2(A) \times D_2(A)$ 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.