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。虽然拓扑变换群很容易通过它们作为对称的对象与几何联系在一起，但对于更一般的群，即配备了乘法或加法概念的集合，情况并非如此。例如，微分方程解通常可以视为无限维空间中的点或向量，即所谓的Banach空间，这些点可以相加在一起形成空间中的新点。所以微分方程解空间本身就是群。事实证明，群体具有内在定义的大规模几何结构。也就是说，当从越来越远的地方观察群体时，微观结构变得模糊，留下一个宏观的视角，可以通过计算获得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各种起源于逻辑、拓扑和分析的无限维群。虽然离散或局部紧群的大规模几何一直是一个非常活跃的领域，但人们最近才意识到波兰群可能具有定义良好的大规模几何。需要研究的几个方面涉及各种具体群类的大规模几何，如紧致流形的同胚群和微分同胚群以及可数一阶结构的自同构群。在后一种情况下，一个特别的问题是用结构的模型理论性质来校准自同构群的几何性质。该提案的性质是非常跨学科的。它一方面涉及到描述集论，通过对波兰群的研究，通过模型论，通过对自同构群的研究，同时融入了几何群论的多方面工具。另一方面，该理论允许对一些迄今不符合这种方法的拓扑变换群进行几何研究，这应该再次打开与几何和微分拓扑学等领域的联系。

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.