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.

人们可以将离散几何简单地描述为对欧几里德空间和非欧几里德空间中几何对象的离散排列的研究。离散几何与纯数学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相关。 Karoly Bezdek博士（加拿大大学）的提议属于离散几何的上述广泛领域，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资深研究人员和初级研究人员以及本科生和研究生的联合合作，促进几何学、分析学和组合学之间的相互作用。另一方面，该提案旨在成为本科生、研究生以及博士后的指导和培训的基础。更详细地说，Bezdek 博士提案的研究部分延续了他之前的 NSERC 发现资助的研究工作，涉及球多面体、接触图、软填料、完全可分离填料、圆柱体覆盖凸体以及凸体的不可分离排列等主题，并围绕这些问题提出了一些新的研究问题。另一方面，他计划开展全新的研究项目，例如通过软球堆积结晶、r 球体的马勒型问题、圆柱体堆积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获得的，因此希望他们的解决方案能够推动这些应用并为其创建新的数学理论。此外，该提案针对的是Bang（1951）、修正的Goodman-Goodman猜想（1945）、Kneser-Poulsen猜想（1955）和Gromov猜想（1987）这两个长期存在的离散几何基本问题的猜想。所提出的方法是离散几何、凸几何和微分几何、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范围以及缩小研究与学术教学之间的差距，引进一批新的本科生、研究生以及博士后进行几何研究。由于最近在离散几何方面取得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。