Mathematical Sciences: Non-Positive Curvature, Triangulations and Topology

小行星9505136 几何与拓扑中的一个基本问题是研究曲率（关于空间的局部信息）与拓扑（关于空间的全局信息）之间的关系。 调查研究一个特殊的情况下，曲率：空间的非正曲率，在黎曼的情况下，并在PL（分段线性）的情况下。他还研究了有关Riemannian和PL刚性（在非正弯曲的情况下）的一些问题; Riemannian和PL非正弯曲几何之间的关系;以及拓扑如何确定曲率：哪些空间允许非正弯曲几何。 非正弯曲几何空间是这样一种空间，在这种空间上，物体在远离观察者时，其尺寸似乎以等于或快于“真实的世界”（欧几里得几何）的速度减小。 一般来说，几何决定了空间的（全局）形状的一些属性。 例如，如果一个空间有一个非正弯曲的几何，在“展开”这个空间之后，我们得到另一个空间，它不像甜甜圈，可以在自身内部收缩到一个点（Cartan-Hadamard定理）。 这个研究项目的方向是研究非正几何空间上的这种关系：几何（局部数据）如何决定全局形状（拓扑）以及拓扑如何决定几何。 如果只考虑地球表面上的欧几里德环境，那么这与“真实的世界”的相关性就不明显，但宇宙学问题--整个宇宙的性质，而不仅仅是我们在其中的一个小角落--与这些数学问题密切相关。 ***

9505136 Ontaneda A basic problem in Geometry and Topology is to study the relation between curvature (local information about a space) and topology (global information about a space). The investigator studies a particular case of curvature: spaces of non-positive curvature, in the riemannian case and in the PL (piecewise linear) case. He also studies some problems concerning riemannian and PL rigidity (in the non-positively curved case); the relation between riemannian and PL non-positively curved geometry; and how the topology could determine the curvature: which spaces admit a non-positively curved geometry. A space with non-positively curved geometry is a space on which, essentially, an object, as it moves away from an observer, seems to decrease in size at a rate equal to or faster than the rate in the "real world" (euclidean geometry). In general, geometry determines some properties about the (global) shape of the space. For instance, if a space has a non-positively curved geometry, after "unwrapping" the space we obtain another space that, unlike a doughnut, can be shrunk within itself to a single point (Cartan-Hadamard theorem). This research project is oriented towards studying such relations on spaces with non-positive geometry: how the geometry (local data) determines the global shape (topology) and also how topology could determine the geometry. Although the relevance of this to the "real world" is not obvious if one considers only the apparently Euclidean environs of the earth, questions of cosmology -- the nature of the universe as a whole and not merely our small corner of it -- are intimately related to these mathematical questions. ***