The recognition problem for compact, simply connected contact manifolds of given dimension $2n-1\geq 11$ (Seidel, 2007).
A contact structure on a $(2n-1)$-manifold is a tangent hyperplane field $\xi$ which can locally be written as $\ker\alpha$ for a 1-form $\alpha$ with $\alpha \wedge (d\alpha)^{n-1}$ non-vanishing. In principle there are finite ways to specify contact manifolds, using symplectic handlebody theory. But there's a simply connected contact manifold $(M_0,\xi_0)$ with the property that, given another, say $(M,\xi)$, the problem of deciding whether it's isomorphic to $(M_0,\xi)$ (M_0,\xi_0)$ contains an algorithmically-unsolvable word problem for groups.
If you forget the contact structure, algorithmic reognition recognition is possible (Nabutovsky-Weinberger).
