This project is to study the new partial differential equations arising from complex geometry and the search for a unified theory of all fundamental interactions. These equations present many novel challenging features and are interesting in their own right, so this is a particularly important area at the interface of mathematics and physics. The project also incorporates the training of graduate students and postdocs and organizing seminars and conferences. More specifically, in Kaehler geometry, equations on spaces with singularities will be studied. It is necessary to allow singularities in view of many applications to algebraic geometry and string theory. In non-Kaehler geometry, a key step will be to identify the underlying special geometry in the sense of holonomy, and weakly parabolic flows will play a major role. New tools will be developed, building in particular on the new methods for estimates introduced recently by the PI and his collaborators. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.