Topics in Discrete Geometry

离散几何主题

• 批准号: RGPIN-2014-06423
• 负责人: Bezdek, Karoly
• 金额: $ 2.04万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=645392
• 关键词: Topics; Discrete; Geometry

The search for densest packings of congruent balls has a long and fascinating history in mathematics. Starting with investigations of Kepler and Gauss and continued by many others, the area of research was finally systematically established by the extensive research work of Coxeter, Delone, Fejes Toth, Rogers and Zassenhaus; and as a result the field of Discrete Geometry was born. Thus, one can briefly describe discrete geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. In recent years renewed interest was generated by quite a number of breakthrough results. As a result discrete geometry has very strong connections to a number of research areas in pure mathematics such as convexity (see in particular, theory of convex bodies and polytopes), combinatorics (see for example, geometric graphs), rigidity (see in particular, flexibility of discrete geometric structures), geometric analysis (see for example, geometry of normed spaces), computational geometry (see for example, computation of volume), geometric groups (see for example, symmetries of polytopes), non-Euclidean geometry (see for example, spherical geometry) as well as to some research areas in communication and information technologies (see in particular, spherical codes) and in crystallography (see in particular, geometric lattices). In addition, there is demand from engineering, biology, and computer science for more emphasis on problems that are discrete in nature and do not fit into the usual continuous models. The research topics proposed represent the above concrete connections. On the other hand, the proposed research program is a combination of fundamental problems of discrete geometry that are connected to important problems of convex and non-Euclidean geometry, including the geometric theory of normed spaces with particular aspects in analysis and combinatorics. The timing seems to be perfect for achieving outstanding advances by intensive scientific collaboration, as well as by training a highly selected group of undergraduate and graduate students and postdoctoral fellows. In addition, the Center for Computational and Discrete Geometry within the Department of Mathematics and Statistics at U of C, which has been established and is supported by my Canada Research Chair (Tier 1) program, together with my proposed research program, has the potential to open a window into what research in modern discrete geometry is really like.

寻找全等球的最密堆积在数学领域有着悠久而迷人的历史。从开普勒和高斯的研究开始，并由许多其他人继续研究，通过考克塞特、德隆、费耶斯·托特、罗杰斯和扎森豪斯的广泛研究工作，最终系统地建立了该研究领域；结果，离散几何领域诞生了。因此，我们可以将离散几何简单地描述为对欧几里德空间以及非欧几里德空间中几何对象的离散排列的研究。近年来，大量突破性成果重新引起了人们的兴趣。因此，离散几何与纯数学中的许多研究领域有着非常密切的联系，例如凸性（特别参见凸体和多面体理论）、组合学（例如参见几何图形）、刚性（特别参见，离散几何结构的灵活性）、几何分析（例如，参见规范空间的几何）、计算几何（参见例如体积的计算）、几何群（例如参见多面体的对称性）、非欧几里得几何（参见例如为了例如，球面几何）以及通信和信息技术（特别参见球码）和晶体学（特别参见几何晶格）的一些研究领域。此外，工程、生物学和计算机科学要求更多地强调本质上离散且不适合通常的连续模型的问题。提出的研究课题代表了上述具体联系。另一方面，所提出的研究计划是离散几何基本问题的组合，这些问题与凸几何和非欧几何的重要问题相关，包括规范空间的几何理论以及分析和组合学的特定方面。现在似乎是通过密集的科学合作以及培养一批精心挑选的本科生、研究生和博士后来取得显着进步的完美时机。此外，加州大学数学与统计系内的计算和离散几何中心已建立并得到我的加拿大研究主席（一级）计划以及我提议的研究计划的支持，该中心具有潜力打开一扇了解现代离散几何研究到底是什么样的窗口。