In this talk, I will present a new index theory for manifolds with singularities (such as manifolds with corners and more generally for manifolds with polyhedral type boundary). Applications of this new index theory include a positive solution to Gromov’s dihedral extremality/rigidity conjecture in all dimensions (joint work with Jinmin Wang and Guoliang Yu). This conjecture concerns comparisons of scalar curvature, mean curvature and dihedral angles for compact manifolds with polyhedral type boundary, and has very interesting implications in geometry and mathematical physics. Further developments of this new index theory have led us to a positive solution of Gromov's flat corner domination conjecture (joint work with Jinmin Wang). As a consequence, we answered positively the Stoker conjecture (a long standing conjecture in discrete geometry since 1968).