Geometry and Topology in the Presence of Lower Curvature Bounds

摘要奖：DMS 1209387，首席研究员：卡斯滕·格罗夫这位首席研究员计划继续他在微分几何及其相关领域的全球问题上的工作。将特别强调通过额外的结构追求全球黎曼几何。这包括但不限于由于对称性的存在而产生的结构，以及由于采用外部或内部限制而产生的结构。我们关于曲率、对称性和拓扑之间关系的研究工作是在研究者发起的所谓对称程序的指导下进行的，其目的是：对具有正曲率或非负曲率和大等距群的流形的结构进行分类或描述。在过去的二十年里，在许多人的贡献下，这一领域在不同方向上取得了重大进展，产生了许多分类类型定理，发现了新的结构，并在非负曲率和正曲率的错觉情况下发现了许多例子。最近，刚性现象已经出现，这与建筑的几何形状有关。在这里，通过所谓的房间系统的Hausdorff拓扑来介绍建筑物的拓扑。该项目的主要重点将是进一步发展这种与建筑物的联系，但也将包括对流形在较低曲率范围下坍塌所产生的结构的调查，以及比较几何与与“流形价值数据处理”有关的新应用领域之间的最近联系。后者也通过非线性“重心”和内部极限来处理平均值。该项目涉及经典刚性和最大对称性欧几里得、球面和双曲几何以及曲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，几何是指空间在保持距离变换(这里称为对称)下不变的那些性质，而拓扑是指空间在拉伸、弯曲和变形等变换下不变的更灵活的性质。曲率控制着测地线的局部行为，即空间中“直线”的行为。相比之下，正曲线空间中测地三角形的角度和大于180度，即三角形在平坦的欧氏平面上的角度和。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发挥着至关重要的作用。大多数拟议的活动都属于调查人员设计的“对称计划”的保护伞。在接下来的几年里，主要的具体目标是一方面寻找更多的正曲线和非负曲线流形的新例子，并在相反的极端表明所谓的对称空间确实是自然意义上的刚性对象。后者是基于最近发现的一个非常不同的领域的链接，即建筑的Tits几何，这一领域在数学中有着深刻而多样的应用。其他重要目标包括在新兴的多值数据处理领域的推动下，在几何学领域发展新的方向，例如计算机视觉、医学成像、传感器网络和形状统计分析。

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.