Geometry of contact transformations and domains: orderability versus squeezing

Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius.Does there exist an analogue of this result in contact geometry?Our main finding is that the answer depends on the sizes of the domains in question: We establish contact nonsqueezing on large scales, and show that it disappears on small scales.The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold.In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold.For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable.Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology.We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups.