Mathematical Sciences: Equilvalence Relations Induced by Polish Group Actions

DMS 9622977 Greg Hjorth，UCLA 群的概念是现代数学中的核心概念，特别是作用于集合的群。 这导致了一个轨道等价关系：对于群G，我们有一个点的轨道是所有gx的集合，其中g在G中。这反过来又促使数学家们对轨道等价关系进行抽象的研究，而这种关系现在已经从最初对作用量的研究中分离出来了。 虽然一些研究者在遍历理论边缘的工作阐明了G是局部紧的情况，但逻辑学的分支（称为模型论）密切关注由可数集N的所有置换的无限对称群S诱导的轨道等价关系，并配备了逐点收敛的拓扑。 Hjorth的项目是针对统一这些非常不同的领域下的连续行动的研究可分离度量化拓扑群，俗称波兰集团。 这类群包括一些熟悉的拓扑群，它们不是局部紧的，例如：S，单位区间的同胚群，希尔伯特空间的自同构群。 群体诉讼的概念可以通过下面的简单类比来理解。假设一粒沙子在空间中运动，我们知道它在给定时间的位置和速度，称之为t。原则上，应该可以在稍后的时间计算出同一斑点的位置和速度，称之为t + s。在这个意义上，我们可以说，真实的数s“作用”于所有可能的位置和速度的集合。 从时间t的任何位置和速度，数s产生时间t+s的位置和速度。 从一个指定的起点可以到达的所有可能的状态都叫做轨道。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，而是看它在一年中所能占据的所有位置的集合：换句话说，它的轨道。Hjorth的项目在一个非常普遍的背景下考虑了群体行为和轨道，特别关注从逻辑考虑中产生的复杂群体。 某些分类提供了这些群体的行动和轨道，以了解当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性质的理解，例如空间中点的轨道。

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.