Distributed obstruction finding and fullerene research

所要求的设备包括研究生和本科生的工作站，它将在我们的理论实验室公开使用，以及一些没有监视器的处理器，以便能够对两个研究问题进行广泛的分布式计算。图论中最著名的定理之一是库拉托夫斯基定理：如果一个图不能在没有交线的平面上绘制，那么它一定包含一个与两个图中的一个同胚的子图。我的研究项目正在进行的一个主要目标是为环面(形状像甜甜圈的曲面)找到类似的定理。额外的计算资源将极大地方便搜索这些障碍，最终目标是找到一个计算机辅助证据，证明我们找到的障碍集是完整的。拥有一组完整的障碍物可以用来证明关于环形图的定理，并有助于设计简单、快速的算法来绘制环形图。富勒烯是全碳分子，其结构是面大小为5或6的3正则平面图。我们已经开发了一个名为FuiGui的程序来帮助可视化这些分子及其参数。我们的目标是积累所有小富勒烯的参数信息的大型数据库，以便研究这些参数与富勒烯的重要化学性质之间的相关性。额外的计算设施对于在合理的时间内收集这些数据至关重要。

The equipment requested includes workstations for graduate and undergraduate students which will be publically available in our theory lab and some processors without monitors to enable extensive distributed computations for two research problems.One of the most famous theorem in graph theory is Kuratowski's theorem: if a graph cannot be drawn on a plane with no crossing lines then it must contain a subgraph homeomorphic to one of two graphs. One major ongoing goal of my research program is to find a similar theorem for a torus (a surface shaped like a doughnut). Extra computational resources would greatly facilitate searching for these and the ultimate goal of finding a computer-aided proof that the set of obstructions we have found is complete. Having a complete set of obstructions could be used to prove theorems about toroidal graphs and would give insight as to how to design simple, fast algorithms for drawing pictures of them.Fullerenes are all-carbon molecules whose structures are 3-regular planar graphs with face sizes five or six. We have developed a program, FuiGui to aid in the visualization of these molecules and their parameters. Our goal is to amass a large database of parameter information for all the small fullerenes in order to investigate correlations between these parameters and important chemical properties of fullerenes. Extra computational facilities are critical for collecting this data in a reasonable amount of time.