2. Geodesics on K3 surfaces

Closed geodesics on K3 surfaces
A theorem due to Bourguignon and Yau [MR0334079] states that if there is a locally length minimizing closed geodesic on a $K3$surface or its quotients, then all sectional curvatures must vanish identically along it. More recently, by studying the physics of certain twodimensional nonlinear sigma models, Douglas and Gao [arXiv:1301.1687] have conjectured the existence of locally length minimizing closed geodesics on nonflat compact Calabi–Yau manifolds. The existence, location and index of these is known for specific cases.Problem 2.1.
[Gonçalo Oliveira] To determine the existence, location, and index of closed geodesics on $K3$surfaces. It is known that their curvature tensor does not vanish in large regions, thus imposing a strong geometric constraint on the location of such conjectural closed geodesics. 
Gromov–Witten invariants for K3, ALG or ALH spaces
Problem 2.2.
[Max Zimet] To determine the open string Gromow–Witten invariants for K3 surfaces, ALG, or ALH spaces. The Gaiotto–Moore–Neitzke provides a conjectural formalism for this goal, and is unknown for manifolds with mirrors and mirror symmetry.
Cite this as: AimPL: Geometry and physics of ALX metrics in gauge theory, available at http://aimpl.org/geomphysalx.