Bruhat-Tits Geometry and Nonnegative Curvature

Bruhat-Tits 几何和非负曲率

• 批准号: 1509162
• 负责人: Karsten Grove
• 金额: $ 31.41万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1509162&HistoricalAwards=false
• 关键词: Bruhat; Tits; Geometry; Nonnegative; Curvature

AbstractAward: DMS 1509162, Principal Investigator: Karsten GroveWith ancient roots, Geometry is a vast, diverse and highly developed, yet continually evolving discipline with connections to virtually all areas of Mathematics, the Physical Sciences and engineering, as well as continually emerging applied fields. Riemannian geometry provides a large and flexible extension of the classical rigid and maximally symmetric "Euclidean," "Spherical" and "Hyperbolic geometries," as well as of the theory of surfaces. The special but rich class of symmetric spaces, the closest generalizations of the sphere, Euclidean plane, and hyperbolic plane, are the jewels and cornerstones among all Riemannian spaces. They play significant roles in several other areas of mathematics as well, including Analysis, Algebra and Dynamics. These important objects fall into two (dual) classes referred to as "compact type" and "non-compact type," where the members of the first are "more curved" than flat space and the members of the latter are "less curved" than flat space. A central aim of the work is to gain new geometric and dynamic insights that will single out say the symmetric spaces of compact type among all spaces "curved more than flat space." Specifically, the work seeks to provide a characterization through the presence of special (so-called polar) transformations by symmetries, known to be abundant for symmetric spaces. A simple analog of the kind of characterization sought is illustrated by the striking fact that among all spaces "curved more than the sphere of radius 1," only the sphere (and real projective space) support a reflection (i.e., the space is a mirror image of itself).There is a well-known link, due to Dadok, between so-called polar representations and isotropy representations of symmetric spaces. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible compact topological spherical buildings of rank at least three. Polar actions on general Riemannian manifolds constitute a vast extension of polar representations just maintaining the basic geometric features of the latter (having so-called sections). However, the presence of such an actions on a (1-connected) positively curved manifold forces it to be a symmetric space of rank 1 (up to diffeomorphism), unless there are codimension-one orbits. The aforementioned links provide the frame for proving this strong "rigidity type" result. The primary focus of the work to be done is to describe the much larger class of nonnegatively curved manifold in the presence of a polar action. The ultimate aim is to show that the principal building blocks are symmetric spaces or are dominated by symmetric spaces. A basic method to be employed is that of analyzing, describing and ultimately classifying the associated combinatorial chamber systems and buildings as in the case of positive curvature. In the nonnegative curvature case, however, the buildings of interest are affine and hence in a sense infinite dimensional objects, and some of the key links described above for spherical buildings are not yet established for such Bruhat-Tits buildings. The prospects for establishing the missing links may bring new insights not only to affine (topological) buildings, but also to Kac-Moody groups and infinite dimensional symmetric spaces and representations.

Abstractaward：DMS 1509162，首席研究员：Karsten Grovewith Gotemer，几何是一种广阔，多样，高度发达的，但不断发展的纪律，与几乎所有数学领域，物理科学和工程的联系，以及不断出现的应用领域。 Riemannian的几何形状提供了经典刚性和最大对称的“欧几里得”，“球形”和“双曲线几何形状”以及表面理论的较大而灵活的扩展。特殊但丰富的对称空间，球体的最接近概括，欧几里得平面和双曲平面是所有Riemannian空间中的珠宝和基石。它们在其他几个数学领域也起着重要作用，包括分析，代数和动力学。这些重要的对象分为两个（双）类，称为“紧凑型”和“非压缩类型”，其中第一个成员比平坦的空间“更弯曲”，而后者的成员比平面空间“弯曲”。 这项工作的一个主要目的是获得新的几何和动态见解，这些见解将挑出所有空间中紧凑型类型的对称空间“比平坦的空间更弯曲”。具体而言，该作品试图通过存在特殊（所谓的极地）转换来提供表征，对称性对称空间很丰富。一个惊人的事实说明了一个简单的类似物，即在所有空间中“比半径1的弯曲更弯曲”，只有领域（和真实的投影空间）支持反射（即，空间是自身的镜像）。由于dadok是一个众所周知的链接，由于dadok，在所谓的极地代表和iSSOMPROPTICTIONS之间，是一个众所周知的链接。此外，山雀的理论和伯恩斯 - 斯帕齐尔（Burns-Spatzier）提供了非紧密级别等级类型的不可还能的对称空间之间的联系，至少三个和不可约的紧凑型拓扑球形建筑物至少三个。对普通里曼语流形的极地作用构成了极地表示的广泛延伸，仅维持后者的基本几何特征（所谓的部分）。然而，除非有一个consimension-ensens-One Orbits，否则在（1个连接的）正面弯曲的歧管上存在这种作用，迫使它是等级1的对称空间。上述链接提供了证明这种强大的“刚性类型”结果的框架。要做的工作的主要重点是描述在存在极性作用的情况下，在存在极性的情况下，非弯曲的歧管。最终的目的是表明主要构建块是对称空间或由对称空间主导的。要采用的基本方法是分析，描述和最终将相关组合的腔室系统和建筑物分类为正曲率。但是，在非负曲率情况下，感兴趣的建筑物是仿射的，因此在某种意义上是无限的尺寸对象，并且上面描述的针对球形建筑物的一些关键链接尚未建立用于此类Bruhat-tits建筑物。建立缺失链接的前景不仅可以带来新的见解（拓扑）建筑物，还可以带给Kac-Moody团体和无限维度的对称空间和表示。