Geometric Structures on Manifolds

这个项目研究几何结构的各个方面，特别是流形和orbifold上的真实的投影结构。凸射影结构的一致胖退化给出了射影线性群的局部仿射建筑上的极限度量。对于曲面，一般情况称为混合结构。其中一个项目是理解更高维度的图景，在那里，根据不同方向的增长率，存在着某种维度分层。另一个项目是研究投射结构背景下不同同质结构之间的转换。特别是，我们希望确定图的8个顶点代表瑟斯顿几何，和边缘表示一个过渡内的射影几何。其中一些讨论是用无穷小的非标准分析语言最自然地完成的。对不同维度空间几何性质的研究构成了爱因斯坦相对论和超弦理论的基础。我们把空间的弯曲当作引力来体验。这些都是基于黎曼几何的，黎曼几何是研究不同距离的一种方法。另一方面是射影几何。这源于文艺复兴时期艺术家对透视法的研究，基于直线的概念，但没有距离的概念。这种几何图形在计算机图形学中很有用。从长度和距离中解放出来，增加了灵活性。最近有兴趣发展这方面的拓扑结构和应用到理论物理。

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.