Geometry, topology and number theory of low-dimensional manifolds

提案：dms -9971705首席研究员：Alan reid本提案涉及三维和四维流形的结构。我们感兴趣的是有限体积的所谓双曲流形。在3维中，我们的一些主要目的是研究双曲3-流形中某些曲面的存在性和性质。提案的其他部分处理数论与双曲3流形代数之间的相互作用。以前的工作已经表明，从与双曲3流形相关的代数数据中产生的某些不变量如何与3流形的几何和拓扑相关。在四维中，我们将探讨双曲四维流形的构造以及三维流形如何融入其中。这与宇宙学中的问题密切相关。流形是数学和物理学中许多地方自然出现的基本对象。目前对三维和四维流形及其相互作用的研究是基础性的。例如，最近，以双曲空间为模型的三维和四维流形理论已经成为理解宇宙空间几何的核心。我们的工作希望揭示更多关于这些空间的信息。我们在学习中使用的一个主要工具来自数论。就像费马大定理的解一样，数论和我们的双曲空间之间的相互作用是重要的。我们的目的是探索这些联系。

Proposal: DMS-9971705Principal Investigator: Alan ReidThis proposal is concerned with the structure of 3- and 4-dimensional manifolds. Our interests are in so-called hyperbolic manifolds of finite volume. In dimension 3 some of our main aims are to study the existence and properties of certain surfaces in hyperbolic 3-manifolds. Other parts of the proposal deal with interactions between number theory and algebra with hyperbolic 3-manifolds. Previous work has shown how certain invariants arising from algebraic data associated to a hyperbolic 3-manifold are related to the geometry and topology of the 3-manifold. In dimension 4 we will explore constructions of hyperbolic 4- manifolds and how 3-dimensional manifolds can immerse in them. This is closely connected to problems in cosmology.Manifolds are fundamental objects which appear naturally in many places in mathematics and physics. At present the study of 3 and 4 dimensional manifolds and the interplay between them is fundamental. For example, recently, the theory of 3 and 4 dimensional manifolds modeled on hyperbolic spaces has become central to understanding the spatial geometry of the Universe. Our work hopes to reveal more about these spaces. A major tool that we use in our studies comes from number theory. As in the case of the solution of Fermat's Last Theorem, the interaction between number theory and our hyperbolic spaces is significant. It is our intention to explore these connections.