|Geometry of contact transformations: orderability vs. squeezing|
Abstract: I will discuss in the talk two tightly linked problems: analogues of symplectic non-squeezing in contact geometry and existence of a non-trivial partial order on the group of contact transformations. It turns out that the non-squeezing results in contact geometry exhibit a quantum character: they hold only in a large scale. This is related to a quite amazing fact. The group of contactomorphisms of S2n-1 for n≥1 admits a contractible loop generated by a positive Hamiltonian. On the other hand, such loops do not exist for large classes of contact manifolds, e.g. spaces of contact elements. This is joint work with S.-S. Kim and L. Polterovich.