Large scale geometry of Polish groups

拓扑变换群是一个几何对象的对称的集合，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。虽然拓扑变换群通过它们对称的对象与几何非常平凡地联系在一起，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看作是无限维空间中的点或向量，即所谓的巴拿赫空间，这些点可以加在一起形成空间中的新点。微分方程的解空间本身就是群。事实证明，群体具有内在定义的大规模几何结构。也就是说，随着群体从越来越远的地方被观察，微观结构变得模糊，只剩下一个宏观的视角，这个视角可以用于计算，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来研究群，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打算对波兰群体的大规模几何结构进行研究。虽然离散或局部紧群的大规模几何在相当长一段时间内一直是一个非常活跃的领域，但波兰群可能具有定义良好的大规模几何是最近才认识到的。研究的几个方面涉及各种具体类群的大尺度几何，如紧流形的同胚群和差胚群以及可数一阶结构的自同构群。在后一种情况下，一个特别的问题是校准自同构群的几何性质与结构的模型理论性质。这个提议的本质是跨学科的。一方面，它将涉及描述性集合论，通过对波兰群的研究，模型论，通过对自同构群的研究，同时结合几何群论的多面工具。另一方面，该理论允许对一些迄今为止不适合这种方法的拓扑变换群进行几何研究，这再次打开了与几何和微分拓扑等领域的联系。

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.