Gauge Theory and Trivalent Graphs in Three-Manifolds

三流形中的规范理论和三价图

基本信息

批准号：
1808794
负责人：
Tomasz Mrowka
金额：
$ 25.4万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1808794&HistoricalAwards=false
关键词：
Gauge Theory Trivalent Graphs Three

项目摘要

In this research project, the principal investigator will continue work that aims to provide a human-readable proof of the four color map theorem. The four color map theorem was first proved in 1976 by Appel and Haken; the proof involved a huge amount of case checking, so much so that the only feasible method of attack was via computer. This machine-assisted proof, however, cannot be directly checked by a human. Although there have been some subsequent simplifications and clarifications, the general brute-force approach has seemed to be the only line of attack. The four color theorem can be rephrased, though, as a question about the topology of three-dimensional manifolds, and in particular new tools inspired by high energy physics give novel ways to approach a conceptual proof of the four color theorem. This project explores and develops these new insights.The principal investigator and collaborator define a version of instanton Floer homology for trivalent graphs embedded in three-manifolds. This theory enables them to place the four color map theorem firmly in the context of gauge-theory-inspired invariants of three-manifolds. They have proved a fundamental non-vanishing theorem for this Floer homology group, reducing the four color map theorem to a question about computing this invariant for general planar graphs. They are developing new tools for the computation of these and related invariants, and they have discovered a spectral sequence relevant to this computation. The collapse of the spectral sequence for planar graphs would imply the four color theorem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在该研究项目中，主要研究者将继续工作，旨在提供四种颜色图定理的人文可读证明。四个颜色地图定理首先由Appel和Haken于1976年证明；证明涉及大量的病例检查，以至于唯一可行的攻击方法是通过计算机。但是，这种机器辅助的证据不能由人类直接检查。尽管随后进行了一些简化和澄清，但一般的蛮力方法似乎是唯一的攻击路线。但是，可以改写四种颜色定理，以作为有关三维流形的拓扑的问题，特别是受高能量物理启发的新工具提供了新的方法，可以提供新的方法来了解四种颜色定理的概念证明。该项目探讨并开发了这些新见解。该理论使他们能够将四个彩色图定理牢固地放置在仪表理论启发的三个manifolds的背景下。他们证明了该浮点同源性组的基本非变化定理，将四个颜色图定理减少到有关计算此通用平面图不变的问题。他们正在开发用于计算这些和相关不变的新工具，并发现了与此计算相关的光谱序列。平面图的光谱序列的崩溃将暗示四种颜色定理。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。