Foundations and applications of geometric graph theory

图形和网络用于对来自许多领域的结构信息进行建模，例如经济学，工程学，社会科学，遗传学，数学和计算机科学。可视化是帮助人类理解信息的关键工具之一。许多现实世界的应用程序要求图形以易于阅读和理解的方式可视化，或者满足某些物理约束。随着大规模网络的扩散，这一点尤其如此。产生这样的网络布局提出了许多具有挑战性的算法和组合问题。为了在这些问题上取得进展，需要在结构图论中取得深刻的理论成果，以及对几何的深刻理解。该项目的核心是旨在发现图（其拓扑结构和结构）与几何之间的丰富联系，并将其应用于：关于图形可视化的基本问题（如二维和三维图形绘制）;图形在表面上的嵌入;几何图论;以及几何与图论的其他应用，如几何空间。最后，各种图着色问题也将使用两种结构工具以及熵压缩方法的Moser和Tardos的研究

Graphs and Networks are used to model structural information arising from many fields such as economics, engineering, social sciences, genetics, mathematics and computer science. Visualization is one of the key tools that aids humans in understanding information. Many real-world applications require graphs to be visualized in a way that is easy to read and understand, or to be laid out satisfying some physical constraints. This is especially true today with the proliferation of large-scale networks. Producing layouts of such networks presents many challenging algorithmic and combinatorial problems. To make advances on these problems, deep theoretical results in structural graph theory as well as deep understanding of geometry is needed. The core of this project is aimed at discovering rich connections between graphs (their topology and structure) and geometry and applying them to: fundamental problems about graph visualizations (such as graph drawings in two and three dimensions); embeddings of graphs on surfaces; geometric graph theory; and other applications where geometry meets graph theory such as geometric spanners. Finally, various graph colouring problems will also be studied using both structural tools as well as the entropy compression method of Moser and Tardos