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.

属性（t）指出，一组对希尔伯特空间的每个等法动作都有一个固定点，是刚性研究中必不可少的工具。对希尔伯特空间的等距动作可以用低维的同一个同谋的语言来描述，系数在单一表示中。 Chatterji-Drutu-Haglund的结果表明，中位空间作用足以体现在希尔伯特空间上的等距作用理论。通过该提案，PI将扩展最新的合作，并通过低维共同体的工具来了解希尔伯特太空行动和中位空间上的动作之间的联系。Newton确认宇宙“在各个方向上看起来都一样”。这种宇宙学原理得出的结论是，宇宙的大规模几何形状必须是球形的（缺乏平行线），欧几里得（平行线的唯一性）或双曲线（平行线的存在和非独特性）。这些几何形状只能大规模理解，因为爱因斯坦的一般相对论理论断言质量在局部扭曲了时空织物。 Gromov的群体理论方法彻底改变了对大规模几何和对称性的研究。组是对象的对称性的集合。 PI的研究与CAT（0）几何形状有关，这是欧几里得和双曲线几何形状之间的混合物。更具体地说，她研究刚性，可以认为可以解决以下问题：1）一个小组可以表示为某个几何对象的对称性集合吗？ 2）如果是这样，那么这种表示是独一无二的吗？刚性的研究本身就是一种数学现象。然而，这可能有一天会导致对我们宇宙的更好理解。