Geometric graphs and applications

几何图形及其应用

• 批准号: 327566472
• 负责人: Professor Dr. Horst Martini
• 金额: --
• 依托单位: Professur Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/327566472?language=en
• 关键词: Geometric; graphs; applications

Geometric graphs are graphs whose vertices and edges are identified with points and (metric) line segments, respectively. Known examples are Delaunay triangulations or 1-skeletons of polytopes. As "geometrization" of an abstract incidence structure, the notion has many applications, in disciplines like optimization ("trees" in location science), discrete geometry (e.g., Erdös-type problems on metrically extremal point sets), convexity (for instance, polytope theory or bodies of constant width), and also in various non-Euclidean geometries (e.g., in general normed spaces). With this "Neuantrag", Prof. H. Martini (Applicant, TU Chemnitz/Germany) applies together with Prof. A. Kamal (Applicant, University teacher at Alquds University, Abu Dis Jerusalem/Palestine), and Prof. Yaakov S. Kupitz (Applicant, Lecturer at the Hebrew University in Jerusalem/Israel) for a trilateral research project referring to geometric graphs and their applications. We have chosen this subject since all of us work in this field already for many years and published jointly related papers. So we developed already a broad mixture of graph-theoretic and geometric methods which is promising in view of reaching really the goals of the described research program. Our already published results mainly refer to basic properties of geometric graphs as well as to applications of them. Now, within our project, we want to stay in such research directions, and within three years we want to write eight joint publications on the following topics. First, regarding fundamental properties of geometric graphs, we will investigate several classes of geometric graphs, which are extremal regarding various properties, where also colorings will play a role. Second, in applied direction we will study finite point sets with extremal diameter graphs (i.e., typical Erdös-type problems), problems in optimization (location science) and generalized partition problems for finite point sets. All these eight papers will be four-authors papers (with M. A. Perles as external coauthor). In the application we also describe the second joint three-years period (planned via "Fortsetzungsantrag"), which will be more of applied nature, with topics like: constructions of metrically extremal point sets in finite-dimensional real Banach spaces and further applications of geometric graphs in location science (i.e., optimization). Based on all this we hope that, with a deeper study of geometric graphs, our research project will forge links and create new or expand existing interactions between the mathematical fields of discrete and computational geometry, combinatorics and graph theory, convexity, as well as Minkowski geometry. Furthermore, the planned project will positively influence the work of our research groups (for example, lectures in research seminars when visiting each other, and refereeing activities regarding PhD students, with respect to all three universities).

几何图是其顶点和边分别用点和（度量）线段标识的图。已知的例子是Delaunay三角剖分或多面体的1-骨架。作为抽象关联结构的“几何化”，该概念具有许多应用，在诸如优化（位置科学中的“树”）、离散几何（例如，度量极值点集上的Erdös型问题），凸性（例如，多面体理论或恒定宽度的物体），以及各种非欧几里德几何（例如，在一般赋范空间中）。有了这个“新特拉克”，H。Martini（申请人，TU切姆尼茨/德国）与A. Kamal（申请人，巴勒斯坦阿布迪斯耶路撒冷Alquds大学教师）和Yaakov S. Kupitz（申请人，以色列耶路撒冷希伯来大学讲师）的一个三边研究项目，涉及几何图形及其应用。我们之所以选择这个课题，是因为我们都在这个领域工作多年，并共同发表了相关论文。因此，我们已经开发了一个广泛的混合图论和几何方法，这是有希望的，在真正达到所描述的研究计划的目标。我们已经发表的结果主要涉及几何图的基本性质以及它们的应用。现在，在我们的项目中，我们希望保持这样的研究方向，并在三年内就以下主题撰写八篇联合出版物。首先，关于几何图的基本性质，我们将研究几类几何图，它们在各种性质上都是极值，其中着色也将发挥作用。其次，在应用方向上，我们将研究具有极值直径图的有限点集（即，典型的Erdös型问题），优化问题（位置科学）和有限点集的广义划分问题。所有这八篇论文都是四作者论文（M。A. Perles作为外部合著者）。在申请中，我们还描述了第二个联合三年期（计划通过“Fortsetzungsantrag”），这将是更多的应用性质，与主题一样：有限维真实的Banach空间中度量极值点集的构造和几何图形在位置科学中的进一步应用（即，优化）。基于这一切，我们希望，随着对几何图形的深入研究，我们的研究项目将建立联系，创造新的或扩大离散和计算几何，组合数学和图论，凸性以及闵可夫斯基几何的数学领域之间现有的相互作用。此外，计划中的项目将对我们研究小组的工作产生积极影响（例如，相互访问时的研究研讨会演讲，以及关于所有三所大学的博士生的裁判活动）。