Low Dimensional Cohomology and the Geometry of Hilbert Space

性质（T）是研究刚性不可缺少的工具，它指出Hilbert空间上群的等距作用都有一个不动点。希尔伯特空间上的等距作用可以用低维上同调的语言来描述，其系数在酉表示中。Chatterji-Drutu-Haglund的一个结果表明，中值空间作用量的丰富性足以体现Hilbert空间上的等距作用量理论.通过这个提议，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量和中值空间作用量之间的联系。牛顿肯定宇宙“在每个方向上看起来都是一样的。这个宇宙学原理导致了这样的结论：宇宙的大尺度几何必须是球形的（没有平行线），欧几里得的（平行线的唯一性），或双曲的（平行线的存在和非唯一性）。这些几何只能在大尺度上理解，因为爱因斯坦的广义相对论断言，质量局部扭曲时空结构。了解研究大规模几何和对称性是革命性的格罗莫夫的方法，以群论。群是对象的对称性的集合。PI的研究涉及CAT（0）几何，这是欧几里德几何和双曲几何的混合。更具体地说，她研究刚性，这可以被认为是为了解决以下问题：1）一个群可以被表示为某个几何对象的对称性的集合吗？2)如果是这样的话，这种代表性是独一无二的吗？刚性作为一种数学现象，其研究本身就很重要。尽管如此，总有一天它会让我们更好地了解我们的宇宙。

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.