Liouville symmetry groups and pseudo-isotopies

Even though Cⁿ is the most basic symplectic manifold, when n>2 its compactly supported symplectomorphism group remains mysterious. For instance, we do not know if it is connected. To understand it better, one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group of a contact manifold, important for relating (for instance) compactly supported symplectomorphisms of Cⁿ, and contacomorphisms of the sphere at infinity. After explaining this background, the talk will focus on a new result: that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.