This thesis studies some problems in geometry and analysis with techniques developed from non-linear partial differential equations, variational calculus, geometric measure theory and topology. It consists of three independent parts: Chapter I. We study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at most n-2, where n is the dimension of the domain. We will also give an example of an energy minimizing map from a surface to a surface that has a singular point. Thus the n-2 dimension estimate is optimal, in contrast to the n-3 dimension estimate of Schoen-Uhlenbeck (SU) for compact targets. Chapter II. Here we study a new intersection homology theory for currents on a space X with cone-like singularities. This homology is given by a new mass functional M\sbp associated with the perversity index p. For X, it pairs with the intersection homology of Goresky-MacPherson, as well as the L\sp2-cohomology of J. Cheeger. We also give a deformation theorem and then prove the existence of M\sbp-minimizing currents in a given intersection homology class. Chapter III. We construct a weak solution for the heat flow associated with various quasiconvex functionals into homogeneous spaces, in particular, the p-harmonic map heat flow for any p>1. Our proof generalizes previous works (CHN), (CH2) which treated the case for p≥2 where the target is a sphere.