turkmath.org

In this first general talk I will explain what "symplectic" means, and sketch a proof of Gromov's non-squeezing theorem and of Gromov's   2-ball theorem. These basic symplectic rigidity results already have applications to   problems in dynamics, such as short-time super-recurrence and the non-existence of local attractors of certain Hamiltonian PDEs. For the second part, write   $B^4(a)$ for the ball of capacity $a=\pi r^2$, and $Z^4$ for the symplectic   cylinder $D^2(1)\times\mathbb R^2$ where $D^2(1)$  is the disc of area 1. Going beyond Gromov's non squeezing theorem, Sackel, Song, Varolgunes, and Zhu recently showed that for $a>1$ the complement $B^4(a) - S$ of a subset $S$ in the ball cannot   be embedded symplectically into $Z^4$ if the Minkowski dimension of $S$ is $<2$. They also found that this result is sharp provided that $a<2$ and then Brendel extended this to $a<3$. In joint work with Emmanuel Opshtein, we find in any ball $B^4(a)$ a finite union of planar Lagrangian discs $S$ such that $B^4(a)\setminus S$  symplectically embeds into $Z^4$. Among the applications are: capacity killing; non-displaceability of the Clifford torus $T(1/d,1/d)$ from $S$ in $B^4(d)$; and the existence of very short Reeb chords from a Legendrian knot back to itself or to $S$.