Geometry and Topology in the Presence of Lower Curvature Bounds

获奖：DMS 1209387，首席研究员：Karsten grove首席研究员计划继续其在微分几何及相关领域的全球性问题的研究。特别强调将致力于通过附加结构追求全局黎曼几何。这包括但不限于由于存在对称性而产生的结构，以及由于采取外部或内部限制而产生的结构。我们关于曲率、对称和拓扑之间关系的研究工作是由研究者发起的所谓对称程序指导的，该程序的目的是：“分类或描述具有正曲率或非负曲率和大等长群的流形的结构”。在过去的二十年里，在许多人的贡献下，这个领域在各个方向上都取得了重大进展，产生了许多分类类型定理，发现了新的结构和许多非负曲率的例子，以及一个虚幻的正曲率的例子。最近出现了刚性现象，将其与建筑物的几何形状联系起来。在这里，建筑的拓扑结构是通过所谓的腔室系统的豪斯多夫拓扑结构引入的。该项目的主要重点将是进一步发展这种与建筑物的联系，但也将包括对低曲率边界下流形倒塌产生的结构的研究，以及最近比较几何与“流形值数据处理”的新应用领域之间的联系。后者还通过非线性“质心”和内部限制处理平均值。该项目涉及经典刚性和最大对称欧几里得几何、球面几何和双曲几何以及表面理论的广泛而灵活的扩展。寻找和展示几何和拓扑之间的关系是这门学科的核心。这里几何指的是空间在距离保持变换下不变的那些性质（这里称为对称性），而拓扑指的是空间在拉伸、弯曲和变形等变换下不变的更灵活的性质。曲率支配着测地线的局部行为，即空间中“直线”的局部行为。相比之下，正弯曲空间中测地线三角形的角和大于180度，即平坦欧几里得平面上三角形的角和。这种一般类型的几何在许多数学，物理以及最近在信号处理等方面的应用中起着至关重要的作用。大多数被提议的活动都在研究者设计的“对称程序”的保护伞下。在接下来的几年里，主要的具体目标一方面是寻找更多的正和非负弯曲流形的新例子，另一方面是证明所谓的对称空间在自然意义上确实是刚性物体。后者是基于最近发现的与一个非常不同的领域的联系，即它的建筑几何，这个领域在数学中有着深刻而多样的应用。其他重要的目标包括发展几何的新方向，这是由流形值数据处理的新兴领域推动的，例如计算机视觉、医学想象、传感器网络和形状的统计分析。

AbstractAward: DMS 1209387, Principal Investigator: Karsten GroveThe principal investigator plans to continue his work on global problems in differential geometry and related areas. Special emphasis will be devoted to the pursuit of global Riemannian geometry via additional structures. This includes but is not limited to structures arising from the presence of symmetries, and to structures arising from taking external or internal limits. Our efforts concerning investigations of relations between curvature, symmetry and topology is guided by the so-called symmetry program initiated by the investigator and set forth by the aim: "Classify or describe the structure of manifolds with positive or nonnegative curvature and large isometry groups". This area has experienced significant advances in various directions during the past two decades with contributions from many people, resulting in a number of classification type theorems, the discovery of new structures and numerous examples in non-negative curvature and one in the illusive case of positive curvature. Most recently rigidity phenomena providing a link to Tits geometry of buildings has emerged. Here topology of buildings is introduced via the Hausdorff topology of so-called chamber systems. The main focus of the project will be to further develop this connection to buildings, but will also include investigations of structures arising from the collapse of manifolds under a lower curvature bound, and recent connections between comparison geometry and a new applied area concerned with the "processing of manifold-valued data". The latter also deals with averages via non-linear "center of mass" and taking internal limits.The project deals with a vast and flexible extension of the classical rigid and maximally symmetric euclidean, spherical and hyperbolic geometries, as well as of the theory of surfaces. Finding and exhibiting relations between geometry and topology is at the heart of the subject. Here geometry refers to those properties of a space that are invariant under distance preserving transformations (called symmetries here), whereas topology refers to the more flexible properties of a space that are invariant under transformations such as stretching, bending and deforming. Curvature governs the local behavior of geodesics, i.e., of the "straight lines" in the space. By comparison, the angle sum of a geodesic triangle in a positively curved space is bigger than 180 degrees, which is the angle sum of a triangle in the flat Euclidean plane. This general type of geometry plays a vital role in much of mathematics, physics and more recently in applications to signal processing and more. Most of the proposed activity falls under the umbrella of the "symmetry program" designed by the investigator. The main specific goals within the next few years are on the one hand to find additional new examples of positively and nonnegatively curved manifolds, and at the opposite extreme to show that the so-called symmetric spaces indeed are rigid objects in a natural sense. The latter is based on a recently discovered link to a very different area, namely Tits geometry of buildings, an area that has had profound and diverse applications within mathematics. Other important goals include developing new directions in geometry motivated by the emerging field of processing of manifold-valued data to, e.g., computer vision, medical imagining, sensor networks, and statistical analysis of shapes.