Geometry and Topology in the Presence of Lower Curvature Bounds

存在较低曲率界的几何和拓扑

• 批准号: 1209387
• 负责人: Karsten Grove
• 金额: $ 32.03万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1209387&HistoricalAwards=false
• 关键词: Geometry; Topology; Presence; Lower; Curvature

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.

摘要奖项：DMS 1209387，首席研究员：Karsten Grove 首席研究员计划继续在微分几何及相关领域的全球问题上开展工作。将特别强调通过附加结构追求全局黎曼几何。 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。我们在曲率、对称性和拓扑之间关系的研究方面的努力受到研究者发起的所谓对称程序的指导，其目标是：“分类或描述具有正或非负曲率和大等距群的流形的结构” 。在过去的二十年里，在许多人的贡献下，这一领域在各个方向上取得了重大进展，产生了许多分类类型定理、新结构的发现和大量非负曲率的例子，以及一个虚幻的正曲率例子。曲率。最近出现的刚性现象提供了与建筑物的山雀几何形状的联系。这里，建筑物的拓扑是通过所谓的室系统的豪斯多夫拓扑引入的。该项目的主要重点将是进一步发展与建筑物的这种联系，但也将包括对较低曲率边界下流形崩溃所产生的结构的研究，以及比较几何和与“多值数据的处理”。后者还通过非线性“质心”并采用内部极限来处理平均值。该项目涉及经典刚性和最大对称欧几里德、球面和双曲几何以及理论的广泛而灵活的扩展表面。寻找并展示几何和拓扑之间的关系是该主题的核心。这里，几何是指在距离保持变换（这里称为对称性）下不变的空间属性，而拓扑是指在拉伸、弯曲和变形等变换下不变的空间更灵活的属性。曲率控制测地线的局部行为，即空间中的“直线”。相比之下，正弯曲空间中的测地三角形的角和大于180度，这是平面欧几里得平面中的三角形的角和。这种一般类型的几何在数学、物理学以及最近的信号处理等应用中发挥着至关重要的作用。大多数拟议的活动都属于研究者设计的“对称计划”的范畴。未来几年的主要具体目标一方面是找到更多正弯曲流形和非负弯曲流形的新例子，另一方面则表明所谓的对称空间确实是自然意义上的刚性物体。后者基于最近发现的与一个非常不同的领域的联系，即建筑物的山雀几何，这个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（例如计算机视觉、医学成像、传感器网络和形状统计分析）的推动下，开发几何学的新方向。