Geometric Structures in Topology, Geometry, Global Analysis and Dynamics

Following Thurston’s geometrisation picture in dimension three, we study geometric manifolds in a more general setting in arbitrary dimensions, with respect to the following problems: (1) The existence of maps of non-zero degree (domination relation or Gromov’s order); (2) The Gromov-Thurston monotonicity problem for numerical homotopy invariants with respect to the domination relation; (3) The existence of Anosov diffeomorphisms (Anosov-Smale conjecture).