Symplectic

Two submanifolds $N_1,N_2$ of a manifold intersect cleanly if their intersection is a submanifold, with tangent bundle $T(N_1\cap N_2)=TN_1\cap TN_2$. This then implies that locally, the intersection is modelled by the intersection of linear subspaces. What is the right notion of clean intersection if more than two submanifolds are involved? I will explain that there is a good generalization, and present applications to Lagrangian intersections in symplectic geometry, and to simplicial manifolds. (Joint work with Kain Dineen.)