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Monday, October 30, 2017

**Abstract:** Gromov's Non-Squeezing Theorem is a remarkable result in symplectic geometry. It says that the ball $B^{2n}(r)$ of radius $r$ in ${\mathbb R}^{2n}$ can be symplectically embedded in the "cylinder" $B^2(R)\times {\mathbb R}^{2n-2}$ of radius $R$ only if $r\le R$. The proof uses so called J-complex (pseudoholomorphic) curves. We will describe a construction of J-complex disks based on the classical scheme for solving the Beltrami equation in complex analysis. We will discuss related results and open problems.