Local-global principles for representations of quadratic forms
  We prove the local-global principle holds for the problem of representations of quadratic forms by quadratic forms, in codimension $\geq 7$. The proof uses the ergodic theory of $p$-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety. 
Jordan Ellenberg and Akshay Venkatesh