9626721 Freire The proposed research lies in the general area of Riemannian geometry. Specifically, the investigator wishes to pursue the following three topics: nonlinear partial differential equations of evolution type and their singular solutions; long time solutions to the wave equation; the topology of a manifold to the existence of harmonic differential forms. Partial differential equations arise in many different contexts. In differential geometry, they can appear as so called Euler-Lagrange equations describing certain critical points of the energy functional defined on the space of maps between spaces. Certain solutions of these equations called harmonic maps (and wave maps in mathematical physics) then describe energy-minimizing maps that are often of physical significance.
