Bruhat-Tits Geometry and Nonnegative Curvature

获奖：DMS 1509162，首席研究员：Karsten grove与古老的根源，几何是一个庞大的，多样的，高度发展的，但不断发展的学科与连接几乎所有领域的数学，物理科学和工程，以及不断出现的应用领域。黎曼几何为经典的刚性和最大对称的“欧几里得几何”、“球面几何”和“双曲几何”以及曲面理论提供了一个大而灵活的扩展。特殊而丰富的对称空间，是球面、欧几里得平面和双曲平面的最接近的推广，是所有黎曼空间的宝石和基石。它们在数学的其他几个领域也扮演着重要的角色，包括分析、代数和动力学。这些重要的对象分为两个（对偶）类，称为“紧型”和“非紧型”，其中前者的成员比平坦空间“更弯曲”，后者的成员比平坦空间“更不弯曲”。这项工作的中心目标是获得新的几何和动态见解，这些见解将在所有空间中挑出紧凑型的对称空间“弯曲多于平坦空间”。具体来说，该作品试图通过对称的特殊（所谓的极性）变换的存在来提供表征，已知对称空间中丰富。对于所寻求的这种特征，有一个简单的类比可以用一个惊人的事实来说明，即在所有“曲率大于半径为1的球体”的空间中，只有球体（和实投影空间）支持反射（即空间是自身的镜像）。由于Dadok，在所谓的极性表示和对称空间的各向同性表示之间存在着众所周知的联系。此外，Tits和Burns-Spatzier的理论提供了至少秩为3的非紧致型的不可约对称空间与至少秩为3的不可约紧致拓扑球形建筑之间的联系。一般黎曼流形上的极作用构成了极表示的巨大扩展，只是保持了后者的基本几何特征（具有所谓的截面）。然而，这种作用在（1连通）正弯曲流形上的存在迫使它是秩1的对称空间（直到微分同态），除非存在余维1的轨道。上述环节为证明这种强“刚性型”结果提供了框架。要做的工作的主要焦点是描述在极性作用下更大的一类非负弯曲流形。最终目的是表明主要的构建块是对称空间或由对称空间支配。要采用的基本方法是分析、描述并最终对相关组合室系统和建筑物进行分类，就像在正曲率的情况下一样。然而，在非负曲率的情况下，感兴趣的建筑是仿射的，因此在某种意义上是无限维的物体，而上面描述的球形建筑的一些关键环节尚未建立在这种Bruhat-Tits建筑中。建立缺失环节的前景不仅可以为仿射（拓扑）建筑带来新的见解，还可以为Kac-Moody群和无限维对称空间和表征带来新的见解。

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.