Gauge Theory and Trivalent Graphs in Three-Manifolds

在这个研究项目中，首席研究员将继续致力于为四色图定理提供人类可读的证明。四色映射定理最早由Appel和Haken在1976年证明；证明涉及大量的案例检查，以至于唯一可行的攻击方法是通过计算机。然而，这种机器辅助的证明不能由人类直接检查。虽然后来有一些简化和澄清，但一般的暴力方法似乎是唯一的攻击路线。然而，四色定理可以被重新表述为一个关于三维流形拓扑的问题，特别是受高能物理启发的新工具为四色定理的概念性证明提供了新的方法。这个项目探索和发展了这些新的见解。主要研究者和合作者为嵌入在三流形中的三价图定义了一个版本的瞬时花同调。这一理论使他们能够将四色图定理牢固地置于三流形的规范理论启发的不变量的背景下。他们证明了这个Floer同调群的一个基本的不消失定理，将四色映射定理简化为计算一般平面图的不变量的问题。他们正在开发计算这些不变量和相关不变量的新工具，并且他们已经发现了与此计算相关的谱序列。平面图的光谱序列的崩溃意味着四色定理。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.