I will present several new results on face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. For instance, one of the results is a sharp lower bound on the number of interior edges in terms of the number of interior vertices and relative Betti numbers (i.e., dimensions of relative homology). I'll also sketch some of the commutative algebra tools and techniques used in the proofs. This is joint work with Satoshi Murai.