Gauge theory and spatial graphs

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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.