Orbit Equivalence Relations and Classification Problems

轨道等价关系和分类问题

• 批准号: 0501039
• 负责人: Su Gao
• 金额: --
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-15 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501039&HistoricalAwards=false
• 关键词: Orbit; Equivalence; Relations; Classification; Problems

This project concerns two important aspects of the descriptive set theory of definable equivalence relations. On the theoretical side the PI proposes to study the structures of Polish groups and the dynamics of their actions. The scope of the groups ranges from countable nilpotent groups to the full unitary group. On the application side the PI proposes to study the major open problems concerning classification of mathematical structures arising in analysis, topology and geometry. The main focus, though, will be classification problems that are comparable to the orbit equivalence relations induced by actions of the unitary group. One example of such a problem is the classification of bounded linear operators on a separable complex Hilbert space.Many important open problems in various fields of mathematics ask for satisfactory classification of mathematical objects studied in the fields. These classification problems can often be formalized as definable equivalence relations, sometimes even orbit equivalence relations induced by Polish group actions. A striking theory of definable equivalence relations has been developed in the past 15 years or so and complete understanding of the complexity of many classification problems has thus been obtained. This project seeks further development of the descriptive set theory of definable equivalence relations. It is anticipated that mathematics of different fields be brought together in this foundational framework. For many of the old classification problems the descriptive set theoretic perspective is new and worth investigating.

该项目涉及可定义等价关系的描述性集合论的两个重要方面。在理论方面，PI 建议研究波兰团体的结构及其行动的动态。群的范围从可数幂零群到完全酉群。在应用方面，PI建议研究有关分析、拓扑和几何中出现的数学结构分类的主要开放问题。不过，主要焦点将是与酉群作用引起的轨道等效关系相当的分类问题。此类问题的一个例子是可分离复数希尔伯特空间上有界线性算子的分类。数学各个领域中的许多重要的开放问题都要求对这些领域中研究的数学对象进行令人满意的分类。这些分类问题通常可以形式化为可定义的等价关系，有时甚至是由波兰群体行为引起的轨道等价关系。在过去 15 年左右的时间里，人们已经发展出了一种引人注目的可定义等价关系理论，并且由此获得了对许多分类问题的复杂性的完整理解。该项目寻求进一步发展可定义等价关系的描述性集合论。预计不同领域的数学将汇集在这个基础框架中。对于许多旧的分类问题，描述性集合论的视角是新的，值得研究。