Mathematical Sciences: Equilvalence Relations Induced by Polish Group Actions

数学科学：波兰群行动引发的等价关系

• 批准号: 9622977
• 负责人: Greg Hjorth
• 金额: $ 10.8万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622977&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Equilvalence; Relations; Induced

DMS 9622977 Greg Hjorth, UCLA The notion of group is a central one in modern mathematics, specifically that of a group acting on a set. This induces an orbit equivalence relation : for the group G we have that the orbit of a point is the set of all gx where g is in G . This in turn cues mathematicians to the abstract study of the orbit equivalence relation, now isolated from the original investigation of the action. While work of some researchers on the edge of ergodic theory illuminates the case when G is locally compact, the branch of logic known as model theory is intimately concerned with the orbit equivalence relations induced by the infinite symmetric group S of all permutations of the countable set N, equipped with the topology of pointwise convergence. Hjorth's project is directed towards unifying these very diverse areas under the study of continuous actions of separable metrizable topological groups, commonly known as Polish groups. This class of groups includes some familiar topological groups which fail to be locally compact, such as: S, the homeomorphism group of the unit interval, and the group of automorphisms of Hilbert space. The notion of a group action can be understood by the following simple analogy. Suppose a fleck of sand is travelling through space, and we know its position and velocity at a given time, call it t. In principle it should be possible to calculate the position and velocity of the same fleck at a later time, call it t + s. In this sense we can say that the real number s "acts" on the collection of all possible positions and velocities. From any position and velocity at time t, the number s produces the position and velocity at time t+s. All possible states reachable from a specified starting point is what is called an orbit. The terminology suggests an analogy: rather than looking at the position of a planet at one fixed time, we look instead at the collection of all positions it can occupy throughout its year: in other words, its orbit. Hjorth's project considers group actions and orbits in a very general context, with particular focus on complicated groups which arise from logical considerations. Certain classifications are provided these group actions and orbits, to give an understanding of when one is more complicated than another. This research is foundational. A central goal is a deepened understanding of the subtle nature of many mathematical objects, such as orbits of points in space.

群的概念是现代数学中的一个核心概念，特别是指作用于集合上的群。这引出了一个轨道等价关系：对于G群，我们有一个点的轨道是G在G中的所有gx的集合。这反过来又提示数学家对轨道等价关系的抽象研究，现在从最初的作用研究中分离出来。虽然一些研究人员在遍历理论的边缘上的工作阐明了G是局部紧的情况，但逻辑的一个分支即模型论密切关注可数集合N的所有排列的无限对称群S所引起的轨道等价关系，这些群具有点向收敛的拓扑结构。Hjorth的项目旨在通过研究可分离的可度量拓扑群（通常称为波兰群）的连续作用来统一这些非常多样化的区域。这类群包括一些我们所熟悉的局部不紧的拓扑群，如：S，单位区间的同胚群，Hilbert空间的自同构群。群体行动的概念可以通过以下简单的类比来理解。假定有一粒沙在空间中运动，我们知道它在某一时刻的位置和速度，我们称它为t。原则上，我们可以计算出同一粒沙在以后的时刻的位置和速度，我们称它为t + s。在这种意义上，我们可以说实数s“作用于”所有可能的位置和速度的集合。从t时刻的任何位置和速度，数字s产生t+s时刻的位置和速度。从一个特定的起点出发，所有可能达到的状态都被称为轨道。这个术语暗示了一个类比：我们不是看一颗行星在一个固定时间的位置，而是看它在一年中可能占据的所有位置的集合：换句话说，它的轨道。Hjorth的项目在非常普遍的背景下考虑群体行为和轨道，特别关注从逻辑考虑产生的复杂群体。某些分类提供了这些群体的行动和轨道，让人们了解什么时候一个比另一个更复杂。这项研究是基础性的。一个中心目标是加深对许多数学对象的微妙本质的理解，比如空间中点的轨道。