Low Dimensional Cohomology and the Geometry of Hilbert Space

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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.