Symplectic rigidity of O'Grady's manifolds

Mukai classified all finite symplectic groups of automorphisms of K3 surfaces as possible subgroups of one of the Mathieu groups. Since then, Mukai's theorem has been reproven using lattice theoretical techniques, and extended to higher dimensional hyperkähler manifolds. In two joint works with L. Giovenzana (Loughborough), A. Grossi (Chemnitz) et C. Onorati (Roma Tor Vergata), we studied possible cohomological actions of symplectic automorphisms of finite order on the two sporadic deformation types found by O'Grady in dimension 6 and 10. In particular, we showed that, in dimension 10, all symplectic automorphisms are trivial. In my talk, I will explain the connection between our proof and the sphere packing problem.