Coxeter多面体与四维双曲流形

Coxeter多面体及相关应用和组合、拓扑、Lie理论、代数几何、代数数论等学科有着深刻的联系。本项目以双曲Coxeter多面体作为突破口，一方面研究其本身的拓扑、组合和几何； 另一方面借助其与等距离散子群的天然联系进行流形理论上的探索。具体而言，本项目主要着眼于以下三方面的研究：.(1)探究Seigel问题的子问题，利用双曲Coxeter多面体构造（极）小体积的四维（算术） 双曲闭流形；.(2)完全分类特定组合类型的Coxeter双曲多面体； .(3)估计更加精确的的(算术）紧的和有限体积Coxeter多面体的存在维数上界。

Coxeter polytopes and related applications have profound connections with combinatorics, topology, Lie theory, algebraic geometry, algebraic number theory, and other disciplines. In this project, we will study the topology, combinatorics, and geometry of Coxeter polytopes themselves. Moreover, we will explore the manifold theory by means of the natural connection between the Coxeter polytopes and the isometric discrete subgroups. To be precise, this project mainly focuses on the following three aspects of research:.(1).Explore the subproblems of the Seigel problem, constructing minimum-volume or small-volume four-dimensional (arithmetic) hyperbolic closed manifolds..(2).Accomplish the classification of Coxeter hyperbolic polytopes under certain combination constraints..(3).Estimate more accurate upper bounds of existence dimensions for (arithmetic) compact and finite volume hyperbolic Coxeter polytopes.