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。在上个世纪初，法国数学家庞加莱（Poincare）在研究地球、月球和太阳等三体系统的运动规律时，认识到了牛顿定律的重要性。在过去的二十年里，拓扑学在蛋白质和DNA的打结以及现代高能物理理论等问题上得到了应用。与高维空间的拓扑结构相反，三维空间的拓扑结构特别微妙。图论也有着悠久的历史。它是网络及其连接的数学理论，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三维空间，本项目旨在使用拓扑学技术来研究图论中的问题。拓扑学技术将从许多来源中提取，但特别是从规范理论中提取，这是一个起源于基础物理学的领域。该项目将加深我们对拓扑学及其与其他数学和科学领域的相互作用的理解。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。项目活动将在下列具体领域进行。与t.s. Mrowka合作，PI将开发一个新的空间三价图的瞬子同调。这将使用与图在旋转群中补的基本群的表示相关的规范理论来定义，SO(3)。这个SO(3)瞬子同调将是两个元素域上的有限维向量空间。对于任何无桥的空间三价图，将完成SO(3)瞬子同调的维数总是非零的证明。PI将研究一般平面三价图的SO(3)瞬子同调的维数。我们期望维数总是与图的三边着色数有关。如果前两个目标得以实现，那么每个无桥平面三价图都至少承认一个三边着色，这是该领域的一个主要成果。使用较大规范群（如SU(N)）定义的三价图的瞬子同调理论将作为本项目的一部分进行研究。SU(3)的情况有望在理解SO(3)的瞬子同源性方面发挥作用。将探讨SU(N)瞬子同调与量子不变量的范畴（如Khovanov-Rozansky同调）之间的关系。

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 proteins and 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 of 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 a new instanton homology for spatial trivalent graphs. This will be defined using a gauge theory related to representations of the fundamental group of the graph's complement in the group of rotations, SO(3). This SO(3) instanton homology will be a finite-dimensional vector space over the field of two elements. A proof will be completed, that the dimension of the SO(3) instanton homology is always non-zero, for any bridgeless, spatial trivalent graph. 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. If the previous two goals are achieved, it will follow that every bridgeless, planar trivalent graph admits at least one three-edge-coloring, a major result in the field. Instanton homology theories for trivalent graphs defined using larger gauge groups such as SU(N) will be investigated as part of this project. The SU(3) case is expected to play a role in understanding the SO(3) instanton homology. Relations will be explored, between SU(N) instanton homology and categorifications of quantum invariants, such as Khovanov-Rozansky homology.