Flat affine and projective structures on manifolds

有许多不同的几何：欧几里得、球面、双曲线和其他像时空的特殊几何一样的几何。新的物体可以通过取一个已知的几何空间并将某些变换相关的点粘合在一起来创建。例如，如果你拿带有变换的普通平面：(x，y)-&gt；(x，y+n)，n=0，1，2，...，-1，-2，...将相关的点粘合在一起可以得到一个无限长的圆柱体圆周1。将这些点粘合在一起可以有一些非常有趣的性质：仔细观察，结果看起来像我们开始的空间，但从全局来看，它可能更复杂。我的研究是从时空(2个空间维度，1个时间维度)和相关空间获得的，如双曲平面。我也对新兴的离散微分几何领域感兴趣。这里我们考虑空间的曲率等几何实体，并试图将它们的精髓封装在一组有限的数据中，以便我们可以用计算机生成准确的图像。

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.