Uniruling of symplectic quotients of coisotropic submanifolds

Coisotropic submanifolds arise naturally in symplectic geometry as level sets of moment maps and in algebraic geometry in the context of normal crossing divisors. In examples, the Marsden-Weinstein quotient or (Fano) complete intersections are often uniruled. 
We show that under natural geometric assumptions on a coisotropic submanifold, the symplectic quotient of the coisotropic is always geometrically uniruled. 
I will explain how to assign a Lagrangian and a hypersurface to a fibered, stable coisotropic C. The Lagrangian inherits a fibre bundle structure from C, the hypersurface captures the generalised Reeb dynamics on C. To derive the result, we then adapt and apply techniques from Lagrangian Floer theory and symplectic field theory.
This is joint work with Jonny Evans.