Symplectic and Poisson manifolds provide the setting for conservative classical mechanics (via the study of Hamiltonian systems). Related to them, contact manifolds provide a setting for the study of dissipative systems, among other applications. Jacobi manifolds generalize contact structures, allowing for degeneracies, in a similar way to how Poisson manifolds generalize symplectic manifolds. Hamiltonian systems on Jacobi manifolds are useful to study time-dependent Hamiltonian systems, and evolutionary partial differential equations, among other applications.

This talk will serve as background for a short series of follow-up talks on ongoing work.
In this talk we will see how Jacobi geometry can be seen as 'homogeneous Poisson geometry' - for example, a Jacobi manifold can be described as a Poisson manifold equipped with a homogeneous structure. A similar process, called symplectization, also lets us view contact manifolds as homogeneous symplectic manifolds.

This conceptual point of view has proven fruitful, particularly in describing desingularizations of Jacobi manifolds through contact realizations. In follow-up talks we will see how we can use this approach to construct geometry-preserving numerical integrators for Jacobi Hamiltonian systems.