Selected topics in discrete geometry

离散几何选题：离散几何是一个处于纯数学和计算机科学交界处的年轻而快速发展的领域。离散几何这一术语曾经主要代表包装、覆盖和平铺领域，现在已逐渐发展到包括组合几何、凸多面体以及直线、圆、平面、球体和其他几何对象在平面和更高维度上的排列等领域。此外，离散几何在其他领域也有许多应用，例如在数字通信中。也就是说，通信理论中的一个基本问题是确定多维空间中等球的最稠密堆积。事实上，所提出的研究是一项相当广泛的研究，研究球面空间中最密集的球体填充、称为球多面体的凸多面体的曲线类似物、圆柱体和木板的(部分)覆盖、照明问题以及(凸)台球桌的(短)台球。建议的合作者包括我在卡尔加里大学的研究生(分别是高级本科生)、卡尔加里大学计算和离散几何中心的来访者以及来自加拿大、美国和欧洲机构的主要研究人员。

Selected Topics in Discrete Geometry: Discrete geometry is a young and fast developing area on the border line of pure mathematics and computer science. The phrase discrete geometry, which at one time stood mainly for the areas of packing, covering and tiling, has gradually grown to include such areas as combinatorial geometry, convex polytopes, and arrangements of lines, circles, planes, spheres and other geometric objects in the plane and in higher dimensions. Also, discrete geometry has a number of applications in other fields such as in digital communications. Namely, one of the fundamental questions in communications theory is to determine the densest packing of equal balls in multidimensional space. Actually, the proposed research is a rather broad one investigating densest sphere packings in spherical space, curved analogues of convex polytopes called ball-polyhedra, (partial) coverings by cylinders and planks, illumination problems as well as (short) billiards of (convex) billiard tables. The proposed collaborators include my graduate students (resp., advanced undergraduate students) at the University of Calgary, the visitors of the Center for Computational and Discrete Geometry at the University of Calgary and leading researchers from institutions in Canada, USA and Europe.