Take the 2-minute tour ×  In a symplectic manifold
$(X^{2n},\omega)$, a hypersurface $Y\subset X$ has contact-type
if there is a contact form $\lambda$ such that
$d\lambda=\omega|_Y$. Recall that a contact form is a 1-form
with $\lambda\wedge(d\lambda)^{n-1}>0$, i.e. the opposite of a
foliation. For example, a starshaped hypersurface has contact-
type in $\mathbb{R}^{2n}$, or more generally an
$Y\subset\mathbb{R}^{2n}$ transverse to a Liouville vector field
defined in a neighborhood of $Y$. In particular, any contact
manifold $Y$ is a contact-type hypersurface in a symplectic
manifold (the symplectization $\mathbb{R}\times Y$).  Now it's
nice and useful to consider symplectic manifolds where its
boundary has contact-type. And this can usually be done given a
compact symplectic manifold (cobordism between contact
manifolds).  This leads to the question of whether or not you
can always build such a space. In other words: Given a random
hypersurface in $\mathbb{R}^{2n}$, is it of contact-type? How do
you tell whe