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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612211
• 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.

寻找最密集的全等球的填充在数学中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进行，研究领域最终由考克斯特、德龙、费伊斯·托特、罗杰斯和扎森豪斯的广泛研究工作系统地建立起来；因此，离散几何领域诞生了。因此，我们可以简单地将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。近年来，许多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很强的联系，如凸性（具体见凸体和多面体理论）、组合学（例如见几何图）、刚性（特别是见离散几何结构的灵活性）、几何分析（例如见赋范空间的几何）、计算几何（例如见体积计算）、几何群(例如见多面体的对称性)，非欧几里得几何（例如，球面几何）以及通信和信息技术（特别是球形密码）和晶体学（特别是几何晶格）的一些研究领域。此外，工程、生物学和计算机科学需要更多地强调本质上是离散的、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，提出的研究计划是离散几何的基本问题的组合，这些问题与凸和非欧几里得几何的重要问题有关，包括赋范空间的几何理论与分析和组合学的特定方面。通过密集的科学合作，以及培养一批经过精心挑选的本科生、研究生和博士后，这个时机似乎是实现卓越进步的完美时机。此外，在我的加拿大研究主席（Tier 1）项目的支持下，加拿大大学数学与统计系的计算和离散几何中心，以及我提出的研究计划，有可能为现代离散几何的研究打开一扇窗户。