Large scale geometry of Polish groups

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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.