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.

DMS 9622977 Greg Hjorth，加州大学洛杉矶分校的概念是现代数学中的核心概念，特别是在集合上作用的小组的概念。 这会引起轨道等效性关系：对于g组，我们有一个点的轨道是g中g中的所有gx的集合。这反过来又提示了数学家对轨道等效关系的抽象研究，该研究现在与该动作的原始研究中分离出来。 虽然一些研究人员在沿厄尔及性理论边缘的工作阐明了g局部紧凑时的情况，但所谓的模型理论的逻辑分支与可计数集的所有无限对称群体所诱导的轨道等效关系密切关注n，配备了侧面收敛的拓扑。 Hjorth的项目针对统一这些非常多样化的领域，研究可分离的可分离拓扑群体（通常称为波兰群体）的连续作用。 这类组包括一些熟悉的拓扑组，这些拓扑组未能局部紧凑，例如：S，单位间隔的同构组以及希尔伯特空间的自动形态群。 通过以下简单类比可以理解小组行动的概念。假设一阵沙子正在穿越太空，我们知道它在给定时间的位置和速度，称其为t。原则上，应该有可能在以后的时间计算同一斑点的位置和速度，称其为t + s。从这个意义上讲，我们可以说实际数字s“行为”在所有可能的位置和速度的集合上。 从时间t的任何位置和速度，数字s在时间t+s处产生位置和速度。 从指定的起点可以达到的所有可能状态是所谓的轨道。术语表明了一个类比：我们没有在一个固定时间看一个行星的位置，而是看着它一年中可以占据的所有职位的收集：换句话说，是其轨道。 Hjorth的项目在非常一般的环境中考虑了小组的行动和轨道，特别关注逻辑上的考虑因素而引起的复杂群体。 提供了某些分类，这些分类和轨道是为了了解一个人何时比另一个更复杂。这项研究是基础的。一个核心目标是对许多数学对象的微妙性质（例如太空中的轨道）的细微性质深入了解。