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Rigidité du crochet de Poisson en topologie symplectique Theses and supervised dissertations / 2017-09
This masters thesis is an introduction to C0 rigidity phenomena in symplectic topology. More precisely, the main concern is the C0 rigidity of the Poisson bracket on a symplectic manifold. The elementary notions of symplectic geometry and Hamiltonian dynamics are recalled in the first chapter. The second chapter introduces the Hofer geometry of the group of Hamiltonian diffeomorphisms of a symplectic manifold. Chapter 3 concerns the application of Hofer geometry to the study of functionals defined in terms of the Poisson bracket. The main result, due to Buhovski, Entov and Polterovich, is the lower semi-continuity of the functional which assigns to every pair of functions the uniform norm of their Poisson bracket.