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Gauge Theory and Spatial Graphs

规范理论和空间图

基本信息

批准号：
1707924
负责人：
Peter Kronheimer
金额：
$ 26.72万
依托单位：
Harvard University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1707924&HistoricalAwards=false
关键词：
Gauge Theory Spatial Graphs

项目摘要

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.

这个项目将连接两个领域的现代数学研究：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。它的重要性在上个世纪之交被法国数学家庞加莱认识到，在他对支配三体系统运动的运动定律的研究中，如地球，月球和太阳根据牛顿定律运动。在过去的20年里，拓扑学已经在一些问题中得到应用，如DNA的打结和蛋白质，以及高能物理的现代理论。三维空间的拓扑结构，与高维空间的拓扑结构相反，特别微妙。图论也有很长的历史。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在三维空间中，该项目旨在使用拓扑技术来研究图论中的问题。拓扑技术将从许多来源，但特别是从规范理论，一个领域有其起源于基础物理学。该项目将加深我们对拓扑学及其与数学和科学其他领域的相互作用的理解。同时，该项目将培训研究生，并向该领域的研究人员传播成果。与T. S. Mrowka，PI将开发空间三价图的瞬子同调的性质。这种瞬子同调是在以前的工作中使用规范理论构造的，该规范理论与旋转群SO（3）中图的补图的基本群的表示有关。PI将研究一般平面三价图的SO（3）瞬子同调的维数。期望维数总是与图的三边着色数有关。作为证明的垫脚石，我们将使用更大的群SU（3）来构造瞬子同调的一个变体，并使用不动点理论来比较这两个版本。如果前两个目标得以实现，那么从这项工作和其他工作中可以得出，每个无桥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，四色定理是任何平面地图的区域可以只用四种颜色着色的陈述。四色定理以前只在计算机辅助下被证明，希望这个项目可能因此导致第一个人类可读的证明。