Two step flag manifolds and the Horn conjecture
  We give a simplification of Belkale's geometric proof of the Horn conjecture. Our approach uses the geometry of two-step flag manifolds to explain the occurrence of the Horn inequalities in a very straightforward way. The arguments for both necessity and sufficiency of the Horn inequalities are fairly conceptual when viewed in this framework. We provide examples to illustrate the method of proof. 
Kevin Purbhoo