Arrangements of Convex Bodies - the Discrete Geometric Side

凸体的排列——离散几何边

• 批准号: RGPIN-2019-03954
• 负责人: Bezdek, Karoly
• 金额: $ 1.53万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737274
• 关键词: Arrangements; Convex; Bodies; Discrete; Geometric

One can briefly describe discrete geometry as the study of discrete arrangements of geometric objects in Euclidean as well as in non-Euclidean spaces. Discrete geometry has very strong connections to a number of research areas in pure mathematics such as convexity, combinatorics, rigidity, geometric analysis, computational geometry, and geometric groups. Also, it is connected to some research areas in communication and information technologies and crystallography. The proposal of Dr. Karoly Bezdek (U of C) belongs to the above broad area of discrete geometry and it aims at achieving the following two major goals. On the one hand, the nine proposed research problems intend to advance the interplay between geometry, analysis, and combinatorics via joint collaborations with established and junior researchers as well as undergraduate and graduate students. On the other hand, the proposal intends to form the basis of the mentoring and training of undergraduate and graduate students as well as postdoctoral fellows. In somewhat more details, the research component of Dr. Bezdek's proposal continues the research work of his previous NSERC discovery grant on topics such as ball-polyhedra, contact graphs, soft packings, totally separable packings, covering convex bodies by cylinders, and non-separable arrangements of convex bodies via working on a number of new research problems proposed around them. On the other hand, he plans to work on fundamentally new research projects as well such as crystallization via soft ball packings, Mahler-type problems for r-ball bodies, packing convex bodies by cylinders, and volumetric geometry of molecules. As some of these problems have been obtained from applied problems of crystallography and computational biology there is hope that their solutions will progress those applications and create a new mathematical theory for them. In addition, the proposal targets the two covering conjectures of Bang (1951), the revised Goodman-Goodman conjecture (1945), the Kneser-Poulsen conjecture (1955), and the Gromov conjecture (1987), which are longstanding fundamental problems of discrete geometry. The proposed methods are combinations of methods from discrete, convex, and differential geometry, geometric analysis, and probability. The training component of Dr. Bezdek's proposal intends to bring in a number of new undergraduate and graduate as well as postdoctoral students for geometry research by expanding the boundaries of collaborative research work and by closing the gap between research and academic teaching. Due to recent breakthroughs in discrete geometry and due to recent increase in student enrolment on all levels at U of C the timing seems to be ideal for achieving the above goals.

可以将离散的几何形状简要描述为欧几里得和非欧几里得空间中几何对象的离散排列的研究。离散的几何形状与纯数学的许多研究领域具有非常牢固的联系，例如凸度，组合，刚性，几何分析，计算几何和几何组。此外，它与传播和信息技术和晶体学方面的一些研究领域有关。 Karoly Bezdek博士（C）的提议属于上述离散几何形状的广泛领域，其目的是实现以下两个主要目标。一方面，九个提出的研究问题旨在通过与已建立和初级研究人员以及本科生和研究生的联合合作来推动几何，分析和组合制剂之间的相互作用。另一方面，该提案旨在构成本科生和研究生以及博士后研究员的心理和培训的基础。在更多细节中，Bezdek博士的提案的研究组成部分继续了他以前的NSERC Discovery Discovery Grant关于诸如Ball-Polyhedra，触点图，软包装，完全独立的包装，圆锥形器覆盖凸面的身体以及不可分离的凸面的凸面，并通过围绕核心研究提出了一些新的研究问题的凸面。另一方面，他计划从根本上进行新的研究项目，以及通过软球包装，Mahler型问题的结晶，用于R球体的问题，圆柱体包装凸形的凸面以及分子的体积几何形状。由于其中一些问题是从晶体学和计算生物学的应用问题中获得的，因此希望他们的解决方案能够进步这些应用并为它们创建新的数学理论。此外，该提案针对Bang（1951年）的两项涵盖协议，《修订后的Goodman-Goodman猜想》（1945年），Kneser-Poulsen猜想（1955）和Gromov猜想（1987），它们是离散几何学的基本问题。所提出的方法是从离散，凸和差异几何，几何分析和概率的方法组合。 Bezdek博士提案的培训部分旨在通过扩大协作研究工作的界限，并缩小研究与学术教学之间的差距，从而引入许多新的本科和研究生以及博士后学生进行几何研究。由于最近的离散几何形状的突破以及最近的学生入学率增加了C的所有级别，时间安排似乎是实现上述目标的理想选择。