Mathematical Sciences: Equilvalence Relations Induced by Polish Group Actions

DMS 9622977格雷格·赫约斯，加州大学洛杉矶分校群的概念在现代数学中是一个核心概念，特别是作用在集合上的群的概念。这就引出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。这反过来又将数学家引向对轨道等价关系的抽象研究，而现在这种抽象研究与最初的行为研究是分开的。虽然一些遍历理论的边缘研究者的工作说明了G是局部紧的情况，但逻辑的分支模型理论与可数集N的所有置换的无限对称群S所诱导的轨道等价关系密切相关，具有逐点收敛的拓扑结构。Hjorth的项目旨在通过研究可分可度量化拓扑群(通常称为波兰群)的连续作用来统一这些非常不同的领域。这类群包括一些常见的不是局部紧的拓扑群，如单位区间的S同胚群和Hilbert空间的自同构群。集体诉讼的概念可以通过以下简单的类比来理解。假设有一粒沙子在太空中穿行，我们知道它在某一时刻的位置和速度，称它为t。原则上，以后应该可以计算出同一斑点的位置和速度，称为t S。从这个意义上说，我们可以说实数S对所有可能的位置和速度的集合起作用。从时间t的任何位置和速度，S产生在时间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称为轨道。这个术语给人一种类比：我们不是看一颗行星在一个固定时间的位置，而是看它全年可以占据的所有位置的集合：换句话说，它的轨道。Hjorth的项目在非常一般的背景下考虑群体行动和轨道，特别关注由逻辑考虑产生的复杂群体。这些群体行为和轨道提供了某些分类，以便理解何时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，例如空间点的轨道。

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.