Abstract: 

In this talk I will first recall some important properties of symplectic and contact toric manifolds. Then I will focus on linear plumbings over spheres where at least one self-intersection number is non-negative. I will show that these manifolds admit a structure of a symplectic toric manifold with a concave contact boundary. I will show that the boundary inherits a toric structure and, as an application, I will show how to employ the corresponding moment cone in order to conclude if the contact structure on the boundary of the given plumbing is tight or overtwisted.
This talk is based on a joint work with Jo Nelson, Ana Rechtman, Laura Starkston, Shira Tanny and Lyua Wang.