The Grassmannian G(k,n) is a manifold that parametrizes k-dimensional subspaces of an n-dimensional vector space. The Grassmannian is a central object in geometry. Consequently, it is important to understand intersections of subvarieties of Grassmannians. The problem of understanding these intersections is cloasely related to the theory of symmetric functions and the representation theory of the symmetric group.
In this talk I will describe an effective method for computing intersections of subvarieties of Grassmannians in terms of combinatorial objects called Mondrian tableaux. The algorithm gives a fast way of answering corresponding problems in related fields. I will then explain how one can compute intersections of subvarieties in Flag manifolds using similar ideas.