Diophantine approximation on subspaces of $\mathbb{R}^n$ and dynamics on   homogeneous spaces
  In recent years, the ergodic theory of group actions on homogeneous spaces has played a significant role in the metric theory of Diophantine approximation. We survey some recent developments with special emphasis on Diophantine properties of affine subspaces and their submanifolds. 
Anish Ghosh
math.NT math.DS