Large scale geometry of Polish groups

波兰群体的大尺度几何结构

• 批准号: 1464974
• 负责人: Christian Rosendal
• 金额: $ 29.98万
• 依托单位: University of Illinois at Chicago
• 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: 2015-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1464974&HistoricalAwards=false
• 关键词: Large; scale; geometry; Polish; groups

A topological transformation group is the set of symmetries of a geometric object such as the set of rigid motions of 3-dimensional space, rotations of a planar disc or symmetries of another similar object. While topological transformation groups are quite trivially connected to geometry by the objects of which they are symmetries, this is not so for more general groups, that is, sets equipped with a notion of multiplication or addition. For example, solutions to differential equations can often be seen as points or vectors in infinite-dimensional spaces, namely so called Banach spaces, and such points can be added together to form new points in the space. So the solution spaces to differential equations are themselves groups. As it turns out, groups have an intrinsically defined large scale geometric structure. That is, as groups are seen from farther and farther away, the microscopical structure becomes blurred and one is left with a macroscopical perspective, which may be accessible to computation and provide structural information about the groups themselves. The PI will investigate groups from this large scale perspective, in particular expanding the existing theory to various infinite-dimensional groups originating in logic, topology and analysis.The PI intends to conduct research on the large scale geometry of Polish groups. While large scale geometry of discrete or locally compact groups has been a very active area for quite some time, the realisation that Polish groups may have a well-defined large scale geometry is very recent. Several aspects to be investigated concern the large scale geometry of various concrete classes of groups such as homeomorphism and diffeomorphism groups of compact manifolds and automorphism groups of countable first-order structures. In the latter case, one particular issue is to calibrate geometric properties of the automorphism group with the model theoretical properties of the structure. The nature of the proposal is very much interdisciplinary. On the one hand, it will involve descriptive set theory, via the study of Polish groups, model theory, via the study of automorphism groups, while at the same time incorporating the multifaceted instruments of geometric group theory. On the other hand, the theory allows for a geometric study of a number of topological transformation groups that were not hitherto amenable to such an approach, which again should open up for connections with areas such as geometric and differential topology.

拓扑转换组是几何对象的对称性集合，例如3维空间的刚性运动，平面盘的旋转或另一个类似对象的对称性。虽然拓扑转换组通过它们是对称的对象非常微不足道地连接到几何形状，但对于更一般的组而言，它并非如此，即配备有乘法或添加概念的集合。例如，微分方程的解决方案通常可以看作是无限维空间中的点或向量，即所谓的Banach空间，并且可以将这些点加在一起以在空间中形成新的点。因此，差分方程的解空间本身就是群体。事实证明，组具有本质上定义的大规模几何结构。也就是说，从较远和更远的地方可以看出组，显微镜结构变得模糊，并且一个剩下的宏观透视图，可以通过计算来访问，并提供有关组本身的结构信息。 PI将从这个大规模的角度研究群体，特别是将现有理论扩展到源自逻辑，拓扑和分析的各种无限二维群体。PI打算对波兰群体的大规模几何形状进行研究。尽管很长一段时间以来，虽然离散或局部紧凑的组的大规模几何形状一直是一个非常活跃的区域，但非常新近的意识到波兰人可能具有明确的大规模几何形状。需要研究的几个方面涉及各种混凝土类别的大规模几何形状，例如同构和差异歧管的差异和差异群，以及可计数的一阶结构组的自多态群体。在后一种情况下，一个特殊的问题是用结构的模型理论特性来校准自动形态组的几何特性。该提案的性质是跨学科的。一方面，它将通过对自动形态群体的研究来涉及描述性集理论，通过对波兰群体，模型理论的研究，同时结合了几何群体理论的多方面工具。另一方面，该理论允许对迄今不适合这种方法的许多拓扑转换组进行几何研究，这些拓扑转换组应再次与几何和差异拓扑等领域的联系开放。