From Poisson geometry to Jacobi geometry and beyond
Poisson geometry is an active area with many applications. It is known that Poisson, contact and lcs geometries have a common extension: Jacobi geometry. It is the geometry of Jacobi structures: Lie brackets on sections of a line bundle that are 1st order bidifferential operators (DOs). Although its feature, Jacobi geometry is far less studied than Poisson geometry. The project aims at studying Jacobi structures with tools from Poisson geometry, Lie theory and PDE theory. It is formed by 3 subprojects.
1 Normal forms in Jacobi geometry. Unlike symplectic/contact structures, Poisson/Jacobi structures have local invariants so their geometry is locally nontrivial. Several normal form theorems for Poisson structures have been proved. We aim at proving similar theorems for Jacobi structures.
2 Contact/Jacobi geometry on b-manifolds. Unlike symplectic/contact manifolds, Poisson/Jacobi manifolds have a complicated geometry. In this regard, b-symplectic structures is a class of more tractable Poisson structures on even-dim b-manifolds. Recently b-contact structures were defined as a class of Jacobi structures on odd-dim b-manifolds. We aim at studying their geometry. 3 Multiplicative DOs & jets. Poisson-Lie groups and symplectic groupoids motivated the study of multiplicative structures. So far only multiplicative structures of tensorial type have been studied. Motivated by Jacobi-Lie groups and contact groupoids, we aim at going beyond tensors, studying multiplicative DOs and jets.