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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=714532
• 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.

人们可以简单地将离散几何描述为研究几何对象在欧几里德空间和非欧几里德空间中的离散排列。离散几何与纯数学中的许多研究领域有很强的联系，如凸性、组合学、刚性、几何分析、计算几何和几何群。此外，它还与通信和信息技术以及结晶学的一些研究领域有关。Caroly Bezdek博士(加州大学)的建议属于上述离散几何的广泛领域，旨在实现以下两个主要目标。一方面，提出的九个研究问题旨在通过与知名和初级研究人员以及本科生和研究生的联合合作，促进几何、分析和组合学之间的相互作用。另一方面，该提案打算成为指导和培训本科生和研究生以及博士后研究员的基础。在更多细节上，Bezdek博士提案的研究部分继续了他之前NSERC发现拨款的研究工作，这些主题包括球多面体、接触图、软填充、完全可分离填充、圆柱体覆盖凸体以及凸体的不可分离排列，通过研究围绕凸体提出的一些新的研究问题。另一方面，他还计划开展一些全新的研究项目，如软球包装的结晶、r球体的马勒型问题、圆柱体的凸体包装以及分子的体积几何。由于其中一些问题是从结晶学和计算生物学的应用问题中获得的，他们的解决方案有望推动这些应用的发展，并为它们创造一种新的数学理论。此外，该提案还针对Bang(1951)的两个覆盖猜想、修正的Goodman-Goodman猜想(1945)、Kenser-Poulsen猜想(1955)和Gromov猜想(1987)，它们都是离散几何的长期基本问题。所提出的方法是离散、凸和微分几何、几何分析和概率的方法的组合。贝兹德克博士提议的培训部分旨在通过扩大合作研究工作的边界，缩小研究与学术教学之间的差距，为几何研究引入大量新的本科生、研究生和博士后。由于最近离散几何方面的突破，以及最近加州大学各级学生入学人数的增加，现在似乎是实现上述目标的理想时机。