Flat affine and projective structures on manifolds

有许多不同的几何：欧几里得几何、球面几何、双曲几何和其他类似时空的特殊几何。新的对象可以通过一个已知的几何空间和“粘合”在一起的点相关的某些转换来创建。例如，如果你取普通平面的变换：(x,y)->(x,y+n) n=0,1,2，…，-1,-2，…把相关的点粘合在一起，就得到了一个周长为1的无限长圆柱体。将点粘合在一起可能会有一些非常有趣的特性：仔细观察，结果看起来就像我们开始的空间，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（如双曲平面）中获得的这样的空间。我对新兴的离散微分几何领域也很感兴趣。在这里，我们采用几何实体，如空间的曲率，并试图将其本质封装在有限的数据集中，以便我们可以用计算机生成精确的图像。

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.