Gromov's non-squeezing theorem and pseudoholomorphic discs -

Abstract

In order to understand the geometry of a given symplectic manifold (M, w), one can study how elementary geometric subsets of M, such as balls, are transformed by symplectomorphisms, i.e. diffeomorphisms preserving the symplectic structure w. Although such diffeomorphisms necessarily preserve the volume, M. Gromov proved in 1985 that symplectomorphisms behave in a more rigid way than volume preserving maps by establishing his celebrated non-squeezing theorem; roughly speaking, one cannot deform symplectomorphically a ball to a thin ball in order to squeeze it in a cylinder. Very recently, A. Sukhov and A. Tumanov gave an elegant and self-contained proof of Gromov's non-squeezing theorem based on the theory of attached pseudoholomorphic discs. The main goal of the proposed Master thesis is to study their approach.