Non-squeezing and other rigidity results in LCS geometry

Locally conformally symplectic (LCS) manifolds are generalisations of symplectic manifolds where the 2-form is not closed but instead satisfies the identity dω = η ∧ ω for a closed 1-form η. The study of these manifolds is equivalent to that of symplectic manifolds when η is exact; however, they resemble the behaviour of contact manifolds when η has no zeroes. Using the theory of generating functions for Lagrangians in the twisted cotangent bundle, we define spectral selectors for Hamiltonian LCS diffeomorphisms of S^1 X R^2n X S^1 and S^1 X R^(2n+1) and a LCS capacity for domains in S^1 X R^2n X S^1, thereby giving us a version of the non-squeezing theorem on this manifold. Time permitting, we shall also see how we can define a partial order on the group of compactly supported LCS Hamiltonian diffeomorphisms on S^1 X R^2n X S^1 and S^1 X R^(2n+1) and a bi-invariant metric on the group of compactly supported LCS Hamiltonian diffeomorphisms of S^1 X R^2n X S^1. This is joint work with Mélanie Bertelson and Margherita Sandon.