Complexity of Hofer's Geometry in Some Higher Dimensional Manifolds

Explore the geometric complexity of Hamiltonian diffeomorphisms in higher-dimensional symplectic manifolds through this advanced mathematical lecture from the IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar. Delve into the profound structure of the group of Hamiltonian diffeomorphisms equipped with the Hofer metric, beginning with foundational results by Polterovich and Shelukhin that established geometric complexity for surfaces and their products by demonstrating sparseness of high powers in the metric space. Examine recent breakthroughs by Álvarez-Gavela and collaborators showing quasi-isometric embedding of free groups into Hamiltonian groups of surfaces, revealing large-scale non-commutativity properties. Learn about generalizations to higher-dimensional symplectic manifolds, including surface bundles, through robust obstruction results that prevent Hamiltonian diffeomorphisms from being nth powers or from being embedded in flows. Discover how every asymptotic cone of Hamiltonian groups for these higher-dimensional manifolds contains an embedded free group, advancing understanding of the geometric and algebraic structure of these fundamental objects in symplectic topology.