Gauge Theory and Spatial Graphs

这个项目将连接两个现代数学研究领域：第一个是拓扑学，第二个是图论和网络流。拓扑学是对空间及其连通性的定性研究。上个世纪之交，法国数学家庞加莱在研究地球、月球和太阳等三体系统的运动规律时认识到了它的重要性。三体系统的运动遵循牛顿定律。在过去的二十年里，拓扑学在蛋白质和DNA的打结等问题以及现代高能物理理论中得到了应用。与高维空间的拓扑结构相比，三维空间的拓扑结构具有特殊的微妙之处。图论也有着悠久的历史。它是关于网络及其联系的数学理论，在计算机科学、算法和优化的许多方面都有应用。通过将网络视为嵌入在三维空间中，该项目旨在使用拓扑学中的技术来研究图论中的问题。拓扑技术将来自许多来源，特别是来自规范理论，这是一个起源于基础物理的领域。该项目将加深我们对拓扑学及其与数学和科学的其他领域的互动的理解。同时，该项目将培养研究生并将成果传播给该地区的研究人员。项目活动将在以下具体领域开展。在与T.S.Mrowka的合作中，PI将开发空间三价图的瞬子同调的性质。这个瞬子同调是在以前的工作中使用规范理论构造的，该规范理论与图的补的基本群在旋转群SO(3)中的表示有关。PI将研究一般平面三价图的SO(3)瞬子同调的维度。通常情况下，维度总是与图的三边着色的数目有关。作为证明的垫脚石，我们将使用更大的群SU(3)来构造瞬子同伦的一个变体，并且将使用不动点理论来比较两个版本。如果前两个目标得以实现，那么从这项工作和其他工作可以得出，每个无桥的平面三价图至少允许一个三边着色，这是该领域的一个主要结果，因为它等价于四色定理，即任何平面地图的区域只能使用四种颜色来着色。四色定理以前只在计算机的帮助下得到了证明，因此希望这个项目可以引领我们走向第一个人类可读的证明。

This project will connect two areas of modern research in mathematics: the first is topology, the second is graph theory and network flows. Topology is the qualitative study of space and its connectedness. Its importance was recognized at the turn of the last century by the French mathematician Poincare, during his investigation of the laws of motion that govern the movement of a three-body system such as the Earth, Moon and Sun moving according to Newton's laws. In the past twenty years, topology has seen applications in questions such as the knotting of and proteins DNA, and in modern theories of high-energy physics. The topology of three-dimensional spaces, as opposed to those of higher dimension, is of particular subtlety. Graph theory also has a long history. It is the mathematical theory of networks and their connections, and sees application in many aspects computer science, algorithms and optimization. By viewing networks as embedded in three-dimensional space, this project aims to use techniques from topology to study questions in graph theory. The topological techniques will be drawn from many sources, but particularly from gauge theory, a field having its origins in fundamental physics. The project will deepen our understanding of topology and its interaction with other areas of mathematics and science. At the same time, the project will train graduate students and disseminate results to researchers in the area.The project activity will be in the following specific areas. In collaboration with T. S. Mrowka, the PI will develop properties of an instanton homology for spatial trivalent graphs. This instanton homology was constructed in previous work using a gauge theory related to representations of the fundamental group of the graph's complement in the group of rotations, SO(3). The PI will investigate the dimension of the SO(3) instanton homology for general planar trivalent graphs. It is expected that the dimension is always related to the number of three-edge-colorings of the graph. As a stepping stone towards the proof, an variant of the instanton homlogy will be constructed using the larger group SU(3), and fixed-point theory will be used to compare the two versions. If the previous two goals are achieved, it will follow from this and other work that every bridgeless, planar trivalent graph admits at least one three-edge-coloring, a major result in the field, as it is equivalent to the four-color theorem, which is the statement that the regions of any planar map can be colored using only four colors. The four-color theorem has been proved previously only with computer assistance, and it is hoped that this project might therefore lead the way to the first human-readable proof.