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: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567792
• 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.

在数学中，寻找全等球的密堆积有着悠久而迷人的历史。从开普勒和高斯的调查和许多其他人继续，该领域的研究终于系统地建立了广泛的研究工作的考克斯特，德龙，费杰斯托特，罗杰斯和Zassenhaus;并因此领域的离散几何诞生。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，许多突破性的成果重新引起了人们的兴趣。因此，离散几何与纯数学的许多研究领域有着非常密切的联系，如凸性（见特别是，凸体和多面体理论），组合（例如，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，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，见球面几何）以及通信和信息技术的一些研究领域（特别参见球形编码）和晶体学（特别参见几何晶格）。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，而这些问题不适合通常的连续模型。所提出的研究课题代表了上述具体联系。另一方面，拟议的研究计划是离散几何的基本问题的组合，这些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与分析和组合学的特定方面。通过密集的科学合作，以及通过培养一批经过严格挑选的本科生、研究生和博士后研究员，实现杰出进展的时机似乎是完美的。此外，该中心计算和离散几何在数学和统计学系在加州大学，这已经建立，并由我的加拿大研究主席（第1层）计划的支持，连同我提出的研究计划，有可能打开一个窗口到什么样的研究在现代离散几何是真的喜欢。