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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588994
• 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.

寻找密度最大的重合球体在数学上有着悠久而引人入胜的历史。从开普勒和高斯的研究开始，经过其他许多人的继续，科克塞特、德隆、费耶斯·托斯、罗杰斯和扎森豪斯的广泛研究工作最终系统地建立了这一研究领域，离散几何领域由此诞生。因此，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。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具有非常强的联系，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(例如参见赋范空间的几何)、计算几何(例如参见体积的计算)、几何群(例如参见多面体的对称性)、非欧几里得几何(参见例如球面几何)以及通信和信息技术中的一些研究领域(尤其参见，球形码)和结晶学(特别是几何晶格)。此外，工程学、生物学和计算机科学要求更多地强调本质上离散的问题，这些问题不适合通常的连续模型。提出的研究课题代表了上述具体联系。另一方面，拟议的研究方案是离散几何基本问题的组合，这些基本问题与凸几何和非欧几里德几何的重要问题有关，包括赋范空间的几何理论，在分析和组合学方面具有特殊的方面。通过密集的科学合作，以及通过培训一批精心挑选的本科生、研究生和博士后研究员，这一时机似乎是取得杰出进展的完美时机。此外，加州大学数学与统计系内的计算和离散几何中心已经成立，并得到了我的加拿大研究教席(Tier 1)计划的支持，连同我建议的研究计划，有可能打开一扇窗，让人们了解现代离散几何的研究到底是什么样子。