Poisson Geometry and its Relatives: A Thematic Day at CRM

Poisson geometry is a natural extension of symplectic geometry that encodes Hamiltonian dynamics, Lie algebras, and singular foliations via skew-symmetric bivector fields. In this talk, I will explore a symmetric counterpart of this framework by introducing symmetric Poisson structures, defined as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying a natural compatibility condition. These new geometric structures extend (pseudo-)Riemannian geometry and describe locally geodesically invariant distributions endowed with a metric along them. In particular, they include totally geodesic foliations equipped with a metric and a compatible connection on each leaf. I will explain how symmetric Poisson geometry is closely related to the Patterson–Walker metric, a split-signature metric on the cotangent bundle that is analogous to the canonical symplectic form.
Finally, I will present several examples of symmetric Poisson structures, with special emphasis on the linear case, which is in one-to-one correspondence with real finite-dimensional Jacobi–Jordan algebras.

In this talk, I will discuss recent joint work with Fabio Gironella on Poisson homeomorphisms, defined as $C^0$-limits of Poisson diffeomorphisms. Our aim is to take some first steps towards the development of what one might call “$C^0$-Poisson geometry”, while addressing several questions raised by Joksimović and Mărcuț. Although passing to the $C^0$-category leads to new flexibility phenomena, certain intrinsic rigidity properties persist. We establish such rigidity results for the symplectic foliation and for coisotropic submanifolds. At the same time, flexibility appears through characteristic foliations and the notion of “liftable” Poisson homeomorphisms. Interestingly, this perspective also leads to new invariants for smooth objects (such as coisotropic submanifolds) and smooth invariants (such as Lie subalgebras) obtained from non-smooth objects. In our proofs, an important role is played by “clean intersection points”, a notion of independent interest that we introduce and study in the context of singular and symplectic foliations.

This is based on joint work with M. del Hoyo. We shall first present a general procedure to differentiate higher Lie groupoids and simplicial manifolds. It is based on quotienting out the information of cochains with higher order vanishing on degenerate simplices. This construction extends the classical ones for groups and groupoids, and comes with a van Est map onto infinitesimal cochains. Finally, we explain a generalized van Est isomorphism theorem under suitable connectivity assumptions and some applications.

The question of existence of codimension one symplectic foliations on manifolds played an important role in the development of Foliation Theory: starting with Reeb’s foliation on $S^3$ all the way to Thurston’s complete characterization in terms of the Euler characteristic. The next step after Reeb’s example was Lawson’s foliation on $S^5$ (and other odd dimensional spheres, and all of them at the end). The similar question for symplectic foliations is far from being understood. While Reeb’s on $S^3$ is easily seen to work, Lawson’s on $S^5$ is already very challenging. And even though the case of $S^5$ was answered positively by Mitsumatsu, the construction is rather technical and seemingly ad-hoc. The aim of this talk is to explain that (stable) generalised complex structures can be used to re-do the case of $S^5$ in a, we believe, much more conceptual and transparent way. This is based on joint work with Gil Cavalcanti.