The set theory of Polish groups

罗森达尔建议从描述集合论、模型理论和拓扑动力学的角度对波兰群体进行一般研究。在该项目中，Rosendal将研究这些群的代数性质和拓扑性质，如小指数性质、Bergman性质、同态的自动连续现象和极端适应性。所研究的主要问题在很大程度上是由由可数结构的自同构群重构可数结构的模型理论问题所推动的。在这方面使用的主要工具之一是自同构群的同构的自动连续性，这推动了对波兰群类之间任意同态的自动连续性这一更广泛现象的研究。Rosendal还计划将这些想法应用于研究不可数离散群的拓扑动力学，尤其是与度量紧性上的不动点有关的拓扑动力学。在一个单独的项目中，Rosendal打算继续他与V.Ferenczi在G.GodeFroy关于非Hilbertian Banach空间中非同构子空间的数目的问题上的工作。目前被证明有用的方法本质上是高度集理论的，并已将问题简化为极小空间的情况。可能更多的分析工具将对下一步更有用。与此相关，Rosendal计划重温Gowers的确定性定理，并通过使用集合论工具将其应用扩展到其名义上的解析集合范围之外。这应该会为Banach空间理论提供经典几何考虑所不能获得的工具。此外，Rosendal将继续与B.D.Miller进行另一个正在进行的项目，将Borel变换分类到Kakutani等价的描述性概念。这是一个与遍历理论中的Kakutani等价理论完全平行的项目，但由于对象的性质，所使用的方法完全是描述性集合论，可以追溯到Glimm和Effros在算子代数中的工作。通过强调数理逻辑与其他数学领域的相互联系，罗森达尔希望丰富逻辑本身，并为数理逻辑之外的对象提供新的见解。他的研究将主要应用于泛函分析(Banach空间理论)、拓扑群和遍历理论。

Rosendal proposes to pursue a general study of Polish groups from the point of view of descriptive set theory, model theory and topological dynamics. In the project Rosendal will investigate both algebraic and topological properties of these groups, such as the small index property, the Bergman property, phenomena of automatic continuity of homomorphisms and extreme amenability. The main problems investigated are to a great extent motivated by the model theoretical problem of reconstructing a countable structure from its group of automorphisms. One of the principal tools used in this connection is the automatic continuity of isomorphisms of automorphism groups, which motivated the study of the broader phenomenon of automatic continuity of arbitrary homomorphisms between classes of Polish groups. Rosendal also plans to apply these ideas to study the topological dynamics of uncountable discrete groups most notably in connection with the fixed point on metric compacta property. The study of Polish groups using the very diverse methods of several fields seems likely to promote the further integration of separate knowledge and deeper understanding of the objects considered.In a separate project, Rosendal intends to continue his work with V. Ferenczi on a question of G. Godefroy concerning the number of non-isomorphic subspaces of a non-Hilbertian Banach space. The methods that have proven useful at this moment have been highly set theoretical in nature and have reduced the problem to the case of minimal spaces. Probably more analytical tools will be of greater utility for the next steps. In connection with this, Rosendal plans to revisit Gowers' determinacy theorem and by using set theoretical tools extend its applications beyond its nominal reach of analytic sets. This should provide tools in Banach space theory not obtainable by classical geometric considerations. Also Rosendal will pursue another ongoing project with B.D. Miller of classifying Borel transformations up to a descriptive notion of Kakutani equivalence. This is a completely parallel project to the theory of Kakutani equivalence in ergodic theory, but due to the nature of the objects, the methods used are completely descriptive set theoretical and go back to works of Glimm and Effros in operator algebra. By stressing the interconnections of mathematical logic with other domains of mathematics, Rosendal hopes to enrich both logic itself and provide new insight into objects outside of mathematical logic. The main applications of his research will be in functional analysis (Banach space theory), topological groups, and ergodic theory.