Low Dimensional Cohomology and the Geometry of Hilbert Space

低维上同调和希尔伯特空间的几何

• 批准号: 1312928
• 负责人: Talia Fernos
• 金额: $ 11.6万
• 依托单位: University of North Carolina Greensboro
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312928&HistoricalAwards=false
• 关键词: Low; Dimensional; Cohomology; Geometry; Hilbert

Property (T), which states that every isometric action of a group on Hilbert space has a fixed point, is an indispensable tool in the study of rigidity. Isometric actions on Hilbert space can be described in the language of low-dimensional cohomology with coefficients in a unitary representation. A result of Chatterji-Drutu-Haglund shows that median space actions are sufficiently rich to embody the theory of isometric actions on Hilbert space. Through this proposal, the PI will expand on recent collaborations and look at the connections between isometric Hilbert space actions and actions on median spaces via the tool of low dimensional cohomology.Newton affirmed that the universe "looks the same in every direction." This cosmological principle leads to the conclusion that the large scale geometry of the universe must be spherical (absence of parallel lines), Euclidean (uniqueness of parallel lines), or hyperbolic (existence and non-uniqueness of parallel lines). These geometries can only be understood in the large scale since Einstein's theory of general relativity asserts that masses locally distort the space-time fabric. Understanding the study of large scale geometry and symmetry was revolutionized by Gromov's approach to group theory. A group is the collection of symmetries of an object. The PI's research is concerned with CAT(0) geometry, a mix between Euclidean and hyperbolic geometries. More specifically, she studies rigidity, which can be thought to address the questions: 1) Can a group be represented as a collection of symmetries of a certain geometric object? 2) If so, is such a representation unique? The study of rigidity is important in its own right as a mathematical phenomenon. Nevertheless, it could one day lead to a better understanding of our universe.

性质(T)是研究刚性不可缺少的工具，它指出群在Hilbert空间上的每个等距作用都有一个不动点。Hilbert空间上的等距作用可以用低维上同调的语言来描述，其中系数在么正表示中。Chatterji-Drutu-Haglund的结果表明，中位空间作用足够丰富，足以体现Hilbert空间上的等距作用理论。通过这一提议，PI将扩展最近的合作，并通过低维上同调的工具来研究等距希尔伯特空间作用与中间空间上的作用之间的联系。牛顿肯定宇宙“在每个方向上看起来都是相同的”。这一宇宙学原理得出的结论是，宇宙的大尺度几何必须是球面(没有平行线)、欧几里得(平行线的唯一性)或双曲线(平行线的存在和不唯一)。由于爱因斯坦的广义相对论断言，局部质量扭曲了时空结构，这些几何图形只能在大范围内被理解。格罗莫夫的群论方法彻底改变了人们对大规模几何和对称性研究的理解。群是对象的对称性的集合。PI的研究涉及CAT(0)几何，它是欧几里得几何和双曲几何的混合体。更具体地说，她研究刚性，这可以被认为解决以下问题：1)一个群是否可以表示为某个几何对象的对称性的集合？2)如果是这样的话，这样的表示是唯一的吗？作为一种数学现象，对刚性的研究本身就很重要。然而，有朝一日它可能会让我们更好地理解我们的宇宙。