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.

我们可以简单地把离散几何描述为研究欧几里得空间和非欧几里得空间中几何物体的离散排列。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晶体学的一些研究领域有关。Karoly Bezdek博士（C大学）的建议属于上述广泛的离散几何领域，其目的是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资深和初级研究人员以及本科生和研究生的共同合作，推进几何、分析和组合学之间的相互作用。另一方面，本方案旨在为本科生、研究生和博士后的指导和培养奠定基础。更详细地说，Bezdek博士的提案的研究部分继续了他之前的NSERC发现资助的研究工作，如球多面体、接触图、软填料、完全可分离填料、用圆柱体覆盖凸体以及凸体的不可分离排列，通过对围绕它们提出的一些新研究问题进行研究。另一方面，他计划从事一些全新的研究项目，比如软球填充结晶、r球体的马勒型问题、圆柱体填充凸体以及分子的体积几何。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得到的，因此他们的解决方案有望推动这些应用，并为它们创造一种新的数学理论。此外，该建议针对Bang（1951）的两个覆盖猜想，修正的Goodman-Goodman猜想（1945），Kneser-Poulsen猜想（1955）和Gromov猜想（1987），这是离散几何长期存在的基本问题。所提出的方法是离散、凸和微分几何、几何分析和概率方法的组合。Bezdek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，为几何研究引入一些新的本科生、研究生和博士后。由于最近在离散几何方面的突破，以及最近在加拿大大学各级学生入学率的增加，现在似乎是实现上述目标的理想时机。