On translated points of contactomorphisms (Geometry Day)
Time:14:00 - 15:00 h
Abstract: Contact manifolds are odd-dimensional manifolds endowed with a maximally non-integrable field of hyperplanes, and naturally appear for instance as hypersurfaces or prequantizations of symplectic manifolds. The tangent bundle of any contact manifold can be written as the direct sum of the contact distribution and a direction that can be thought of as degenerate. Translated points of contactomorphisms are fixed points modulo this degenerate direction, and play an important role (similar to the one played in the symplectic case by fixed points of Hamiltonian diffeomorphisms) in some proofs of certain contact rigidity results (for instance, the contact non-squeezing theorem). In my talk I will describe the notion of translated points and a conjecture on their existence (similar to the Arnold conjecture on fixed points of Hamiltonian diffeomorphisms). I will also sketch the construction (still inspired by the analogy with the symplectic case) of a Floer-theoretical invariant to study them.