This project studies various aspects of geometric structures and in particular real projective structures on manifolds and orbifolds. A uniformly fat degeneration of convex projective structures gives a limit metric locally modeled on the affine building for the projective linear group. For surfaces the general case is called a mixed structure. One project is to understand the picture in higher dimensions, where there is some kind of stratification by dimension depending on rates of growth in different directions. Another project is to study transitions between different homogenous structures within the setting of projective structures. In particular we hope to determine the graph with eight vertices representing the Thurston geometries, and edges representing a transition within projective geometry. Some of this discussion is most naturally done in the language of nonstandard analysis using infinitesimals.
The study of geometrical properties of space in various dimensions forms the basis of Einstein's theory of relativity, and the physics of superstring theory. We experience the curvature of space as gravity. This is all based on Riemannian geometry which is a way to study different kinds of distance. On the other hand there is projective geometry. This arose from the study of perspective developed by artists in the Renaissance and is based on the notion of straight line but without a notion of distance. This geometry is useful in computer graphics. Freeing oneself from length and distance introduces added flexibility. Recently there has been interest in developing this in the context of topology and applying it to theoretical physics.
