When compared to symplectic structures, Poisson structures can
have very wild properties. Some can be very flexible with huge
deformation spaces, others can have very pathological automorphism
groups. Many properties are still only poorly understood for general
Poisson manifolds.
In this talk, I will discuss two open questions in Poisson geometry:
Smoothness of the Poisson diffeomorphism group and the Poisson extension
problem. To illustrate some of the phenomena related to these questions,
I will give explicit examples of Poisson structures, all with compact
supports and constructed in an elementary way. The first examples have a
Poisson diffeomorphism group which is not locally path-connected for the
Whitney topology. The other examples are cases in which we can solve the
Poisson extension problem explicitely.
This talk is based on:
[1] "Poisson structures whose Poisson diffeomorphism group is not
locally path-connected", Ioan Marcut, Ann. H. Lebesgue 4 (2021),
1521–1529.
[2] "Poisson structures with compact support", Gil R. Cavalcanti, Ioan
Marcut, arXiv:2209.14016.
