Geometric Structures on Manifolds

这个项目研究几何结构的各个方面，特别是流形和奥布里流形上的实射影结构。凸射影结构的一致脂肪退化给出了射影线性群局部仿射构造物上的极限度量。对于曲面，一般情况称为混合结构。一个项目是在更高的维度上理解图景，根据不同方向的增长速度，按维度进行某种分层。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。特别地，我们希望确定一个图，该图有八个顶点表示瑟斯顿几何，边表示射影几何中的一个过渡。其中一些讨论是最自然地用使用无穷小的非标准分析语言完成的。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物理学的基础。我们把空间的曲率当作重力来体验。这都是基于黎曼几何，这是一种研究不同类型距离的方法。另一方面，还有射影几何。这源于文艺复兴时期艺术家对透视的研究，基于直线的概念，但没有距离的概念。此几何体在计算机图形学中很有用。把自己从长度和距离中解放出来，会带来更大的灵活性。最近，人们有兴趣在拓扑学的背景下发展这一概念，并将其应用于理论物理。

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.