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.

在数学方面，寻找最稠密的球包装有悠久而有趣的历史。从对开普勒和高斯的调查开始，并由许多其他人继续进行，最终由Coxeter，DeLone，Fejes Toth，Rogers和Zassenhaus的广泛研究工作系统地建立。结果，离散几何学领域诞生了。因此，可以将离散的几何形状简要描述为欧几里得和非欧几里得空间中几何对象的离散排列的研究。近年来，有很多突破性的结果产生了兴趣。 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体积），几何组（例如，请参见多型的对称性），非欧国人的几何形状（例如，参见球形几何形状）以及有关通信和信息技术的某些研究领域（特别是参见球形代码），尤其是在晶体学中（请参阅尤其是几何晶格）。此外，工程，生物学和计算机科学的需求更加重视本质上离散并且不适合通常的连续模型的问题。提出的研究主题代表上述具体连接。另一方面，拟议的研究计划是离散几何学的基本问题的结合，这些几何形状与凸和非欧盟几何的重要问题有关，包括具有分析和组合学特定方面的规范空间的几何理论。时机似乎非常适合通过密集的科学合作以及培训一群精选的本科生和研究生和博士后研究员的杰出进步。此外，在C的数学和统计局内的计算和离散几何形状中心已建立并得到了我的加拿大研究主席（Tier 1）计划的支持，以及我建议的研究计划，有可能为现代离散几何学研究中的研究打开一个窗口。