Large scale geometry of Polish groups

拓扑变换群是几何对象的对称性的集合，例如三维空间的刚性运动的集合，平面圆盘的旋转或另一个类似对象的对称性。虽然拓扑变换群与几何的联系是相当平凡的，因为它们是几何的对称对象，但对于更一般的群，即具有乘法或加法概念的集合，情况并非如此。例如，微分方程的解通常可以被看作是无限维空间（即所谓的Banach空间）中的点或向量，并且这些点可以被加在一起以形成空间中的新点。所以微分方程的解空间本身就是群。事实证明，群具有内在定义的大尺度几何结构。也就是说，当从越来越远的地方看到群体时，微观结构变得模糊，留下一个宏观视角，这可以被计算访问并提供关于群体本身的结构信息。PI将从这个大尺度的角度研究群，特别是将现有的理论扩展到起源于逻辑，拓扑和分析的各种无限维群。PI打算对波兰群的大尺度几何进行研究。虽然大规模的几何离散或局部紧凑的群体一直是一个非常活跃的领域相当一段时间，实现波兰群体可能有一个明确的大规模几何是非常近期。要调查的几个方面涉及大规模几何的各种具体类别的群体，如同胚和同胚群的紧凑流形和自同构群的可数一阶结构。在后一种情况下，一个特殊的问题是校准几何性质的自同构群的模型理论性质的结构。该提案的性质是非常跨学科的。一方面，它将涉及描述集理论，通过研究波兰群，模型理论，通过研究自同构群，同时结合几何群论的多方面工具。另一方面，该理论允许几何研究的一些拓扑变换群，迄今不服从这种方法，这应该再次开放的连接领域，如几何和微分拓扑。

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.