Gauge theory and spatial graphs

规范理论和空间图

基本信息

批准号：
1405652
负责人：
Peter Kronheimer
金额：
$ 39.12万
依托单位：
Harvard University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2018-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405652&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 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.

该项目将连接现代数学研究的两个领域：第一个是拓扑，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。上世纪初，法国数学家庞加莱在研究控制三体系统（例如地球、月球和太阳）根据牛顿定律运动的运动定律时认识到了它的重要性。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。与高维空间的拓扑相比，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。拓扑技术的来源有很多，但特别是规范场论，这个领域起源于基础物理学。该项目将加深我们对拓扑及其与数学和科学其他领域相互作用的理解。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。该项目活动将在以下具体领域进行。 PI 将与 T. S. Mrowka 合作，为空间三价图开发一种新的瞬子同源性。这将使用与旋转群 SO(3) 中图的补集的基本群的表示相关的规范理论来定义。该 SO(3) 瞬子同源性将是两个元素域上的有限维向量空间。将完成证明，对于任何无桥空间三价图，SO(3) 瞬子同调的维数始终不为零。 PI 将研究一般平面三价图的 SO(3) 瞬时同调的维数。预计维度总是与图的三边着色的数量相关。如果实现了前两个目标，那么每个无桥平面三价图都允许至少一个三边着色，这是该领域的一项重大成果。作为该项目的一部分，将研究使用较大规范组（例如 SU(N)）定义的三价图的瞬时同调理论。 SU(3) 情况有望在理解 SO(3) 瞬子同源性方面发挥作用。将探索 SU(N) 瞬时同源性和量子不变量分类（例如 Khovanov-Rozansky 同源性）之间的关系。