Schmidt's Subspace Theorem is a higher dimensional generalization of Roth's Theorem on the approximation of algebraic numbers by elements from a given number field. It asserts that the set of solutions in $P^n(K)$ ($K$ number field) of a particular Diophantine inequality is contained in finitely many proper linear subspaces of $P^n(K)$. I would like to discuss quantitative versions giving an explicit upper bound for the number of subspaces, and go into ideas of Faltings that went into the proof.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3472
[3] http://www.mpim-bonn.mpg.de/de/node/4751