The set theory of Polish groups

波兰群的集合论

• 批准号: 0556368
• 负责人: Christian Rosendal
• 金额: $ 12.12万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-05-15 至 2009-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556368&HistoricalAwards=false
• 关键词: set; theory; Polish; groups

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.

罗森达尔（Rosendal）提议从描述性集理论，模型理论和拓扑动态的角度从波兰人群体进行一般研究。在该项目中，玫瑰登录将研究这些群体的代数和拓扑特性，例如小索引特性，伯格曼特性，同构的自动连续性现象和极端的不正常。研究的主要问题在很大程度上是由模型理论问题引起的，该问题是从其自动形态组中重建可数结构的原因。在此连接中使用的主要工具之一是自动形态群体的自动连续性，这激发了研究波兰人类别之间任意同态自动连续性的更广泛现象。 Rosendal还计划应用这些想法，以研究与公制紧凑型属性的固定点有关的无数离散组的拓扑动态。使用多种领域的非常多样化的方法对波兰人群体的研究似乎很可能促进了单独的知识和对所考虑的物体的更深入了解的进一步整合。在一个单独的项目中，Rosendal打算继续与V. ferenczi有关G. godefroy的问题，讨论G. godefroy的问题，这些问题涉及非希尔伯特式宽松空间的非同型式subspace的数量。目前证明有用的方法本质上是高度设定的，并将问题降低为最小空间的情况。对于下一步，可能会有更大的分析工具具有更大的实用性。与此相关的是，Rosendal计划重新访问Gowers的决定性定理，并使用设置的理论工具将其应用程序扩展到了其标称分析集的范围之外。这应该在Banach空间理论中提供工具，而不是经典的几何考虑。 罗森达尔还将与B.D.一起进行另一个正在进行的项目。将Borel转换分类为Kakutani等效的描述性概念的米勒。这是千古理论中kakutani等效理论的一个完全平行的项目，但是由于对象的性质，所使用的方法是完全描述性设置的理论，并返回到操作员代数中Glimm和Effros的作品。通过强调数学逻辑与其他数学领域的互连，Rosendal希望能够丰富逻辑本身，并提供对数学逻辑之外对象的新见解。他的研究的主要应用是在功能分析（Banach空间理论），拓扑组和千古理论中。