Applications of descriptive set theory to functional analysis and topological dynamics

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI打算研究三个不同的主题，涉及描述集理论及其在巴拿赫空间理论和拓扑动力学中的应用。首先，他打算进一步阐述大波兰群的结构理论，特别是可数一阶结构的自同构群。PI在这一领域的主要兴趣是在这类自同构群中可数离散群的一般表示的存在性及其在研究自同构群的动力学和代数结构中的应用。其次，PI将研究J.P.R. Christensen的经典问题，即波兰群之间是否存在普遍可测量的同态是连续的。同样，这个问题与波兰群体的几何和动态特性密切相关，很可能导致在自动连续性研究之外的新技术和有价值的概念。最后，Rosendal将继续与V. Ferenczi合作，研究可分离巴拿赫空间的粗略分类，这是W. T. Gowers在解决齐次空间问题的过程中提出的一个项目。该项目在很大程度上依赖于描述集合论中的拉姆齐理论和博弈论方法，从而通过开发新技术和功能分析来丰富描述集合论，从而为巴纳赫空间理论的一个最核心问题做出贡献。近年来，研究动力系统和描述集理论的研究人员之间的合作和思想交流越来越多。最初，这在很大程度上是由对Borel等价关系的研究促成的，但最近也开始包括拓扑动力学。描述集合论的主要目标在于研究实线的可定义子集，并表明这些子集比任意子集表现得更好。这已经发展成为这种集合的一个大结构理论，它提供了一个坚实的框架，其中大多数数学分析发生，因此，描述性集合理论技术在部分泛函分析中的使用正在不断扩大。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI intends to work on three different topics involving descriptive set theory and its applications in Banach space theory and topological dynamics. Firstly, he intends to further elaborate on the structure theory of large Polish group and, in particular, automorphism groups of countable first order structures. The PI's main interest in this domain is the existence of generic representations of countable discrete groups in such automorphism groups and its uses in investigating the dynamics and algebraic structure of the automorphism group. Secondly, the PI will be working on the classical problem of J.P.R. Christensen of whether any universally measurable homomorphism between Polish groups is continuous. Again, this problem is tightly connected with the geometry and dynamical properties of Polish groups and very likely to leads to new techniques and concepts of value outside of the study of automatic continuity. Finally, Rosendal will be continuing his collaboration with V. Ferenczi on the rough classification of separable Banach spaces, a programme initiated by W. T. Gowers in the course of the solution to the homogeneous space problem. This programme heavily relies on Ramsey theoretical and game theoretical methods from descriptive set theory and thus enriches both descriptive set theory by developing new techniques and functional analysis by contributing to one of the most central problems of Banach space theory. In recent years, there has been a growing collaboration and sharing of ideas between researchers working in dynamical systems and descriptive set theory. Initially, this was largely fostered by work on Borel equivalence relations but has lately also come to include topological dynamics. The principal objectives of descriptive set theory consists in studying the definable subsets of the real line, showing that these are much better behaved than arbitrary subsets. This has developed into a large structure theory of such sets which gives a solid framework within which most of mathematical analysis takes place, and as such, the use of descriptive set theoretical techniques in parts of functional analysis is continuously expanding.