Flat affine and projective structures on manifolds

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There are many different geometries: Euclidean, spherical, hyperbolic and others like the special geometry of spacetime.  New objects can be created by taking a known geometrical space and "gluing" together points related by certain transformations.  For instance, if you take the ordinary plane with the transformations :(x,y)->(x,y+n), n=0,1,2,...,-1,-2,...gluing together related points gives you an infinitely long  cylinder of circumference 1.Gluing points together can have some very interesting properties: looking closely, the result looks like the space we started from, but globally, it may be more complicated.My research is on such spaces obtained from spacetime (2 space dimensions, 1 time dimension) and related spaces, such as the hyperbolic plane.I am also interested in the emerging field of discrete differential geometry.  Here we take geometrical entities such as curvature of a space, and try to encapsulate their essence in a finite set of data, so that we may produce accurate pictures with a computer.