Taut foliations, representations, and the computational complexity of knot genus

该项目涉及拓扑学（形状的数学研究）和计算复杂性（如何使用计算机有效地解决问题）领域。这个项目从研究三维流形开始。“三流形”是一个局部看起来像我们周围的3D空间的空间。例如，想象一个闭合的，可能缠结的绳子的互补（即，一个结）在3D空间内。如果我们可以连续地将一个纽结变形为另一个纽结而不撕裂第一个纽结，我们（拓扑学家）认为这两个纽结是同一个纽结。我们可以在我们自己的DNA结构中发现这种结的自然发生，DNA的拓扑特征（结）反映了它们所属的人的一些遗传特征。从三维流形继续，我们也看看二维流形，称为“表面”，局部看起来像几何平面。球体和甜甜圈（即，具有一个柄的环面）是表面的最简单的示例。如果我们从每个表面上移去一个小圆盘，我们会得到我们所说的“有边界的表面”。这个曲面的“亏格”是它所拥有的句柄的数量。拓扑学家已经知道，一个纽结的一个重要的拓扑特征确实是“纽结属”。我们可以将纽结（K）的亏格定义为边界与K重合的所有可能缠结的可定向曲面之间的最小亏格。确定纽结的属在相当长的一段时间内一直是一个非常困难的问题。阿戈尔，哈斯和瑟斯顿表明，如果我们允许这两个结和周围的三个流形的变化，那么问题的“结属”是“NP完全”。“NP完全”一词值得进一步解释：从网络理论到金融市场再到互联网安全，计算知识中有许多看似不同的问题，从纯数学的角度来看，所有这些问题实际上都是等价的。这意味着，如果我们有这些问题之一的解决方案，那么我们就有了解开所有这些问题的万能钥匙！因此，一个NP完全问题的解决方案是通往计算知识中大量重要答案的门户。这是数学最迷人的魅力之一：我们的工作可能会统一这些遥远的现象。到目前为止，确定纽结属的唯一实用方法涉及所谓的“叶理理论”。该理论的术语是受地理学中分层岩石的启发，它为我们所谈论的这种空间的永恒性和不确定性提供了一个很好的视觉效果。向前移动，我们理解一个三流形的叶理是将三流形划分成表面（称为“leaf”，这个术语是受树叶的启发），这样局部地，表面就像一堆纸一样适合在一起。这里需要注意的是，也可以有无限的表面，我们在这里不讨论。一类特别重要的叶理被称为“绷紧叶理”。直觉上，一个绷紧的叶理有这样的属性，它所有的叶子最小化面积（像'肥皂膜'，这是当两个肥皂泡合并并在它们之间创建一个薄膜时创建的）Agol，Hass和Thurston的工作对于我们理解“P vs. NP问题”也很重要，这是一个困扰计算机科学家几十年的著名问题。P vs. NP被克莱研究所列入百万美元千禧年奖名单，它是公众每天通过万维网使用的数据加密的基础。我提出的项目旨在了解紧叶理和其他相关的概念，并继续阿戈尔，哈斯和瑟斯顿的工作，进一步了解结属问题。该项目将在数学和计算机科学的不同领域之间建立新的桥梁，并可能对DNA的研究以及我们对P与NP问题的理解产生重要的应用。

This project involves the fields of topology (the mathematical study of shapes) and computational complexity (how to solve questions efficiently using a computer). This project starts with the study of three-manifolds. A 'three-manifold' is a space that locally looks like the 3D space surrounding us. For example, imagine the complement of a closed, possibly tangled rope (i.e., a knot) inside 3D space. If we can continuously deform one knot into another one without tearing that first knot, we (topologists) consider the two knots to be the same knot. We can find a natural occurrence of such knots in our very own DNA structure, where the topological features of DNA (knots) reflect some inherited characteristics of the person they belong to. Continuing from three-manifolds, we also look at 2D manifolds, known as 'surfaces', which locally look like geometric planes. Spheres and doughnuts (i.e., a torus, which has one handle) are the simplest examples of surfaces. If we were to remove a small disk from each of these surfaces, we'd get what we refer to as 'a surface with a boundary'. The 'genus' of this surface is the number of handles it has. Topologists have known that an important topological feature of a knot is indeed the 'knot genus'. We can define the genus of a knot (K) as the minimum genus between all, possibly tangled, orientable surfaces whose boundary coincides with K. Determining the genus of a knot has been a very difficult question for quite some time. Agol, Hass and Thurston showed that if we allow both the knot and the ambient three-manifold to vary, then the question of `knot genus' is `NP-complete'. The term 'NP-complete' deserves further explanation: There are many seemingly different questions across computational knowledge, from network theory to financial markets to internet security, all of which are actually equivalent from the pure mathematical angle. This means that if we have the solution to one of these questions, then we have the master key to unlock them all! Therefore, a solution to one NP-complete question is the gateway to a huge list of important answers across computational knowledge. This is one of the most fascinating beauties of mathematics: our work may unify these otherwise distant phenomena. To this date, the only practical way of determining the knot genus involves what is known as the 'theory of foliations'. The theory's terminology is inspired by stratified rocks in geography, which gives a nice visual to the timeless nature and immensity of the kind of space we're talking about. Moving forth, we understand a foliation of a three-manifold to be a partition of the three-manifold into surfaces (called 'leaves', the terminology being inspired by tree leaves), such that locally, the surfaces fit together no different than a stack of papers. The caveat here is that there can be infinite surfaces as well, something that we do not discuss here. A particularly important class of foliations are called `taut foliations'. Intuitively, a taut foliation has the property such that all its leaves minimize the area (like 'soap films', which are created when two soap bubbles merge and create a thin film between them).The work of Agol, Hass and Thurston is also important for our understanding of the `P vs. NP question', a famous one that has puzzled computer scientists for decades. The P vs. NP is on the list of million-dollar Millennium Prizes by the Clay Institute, and it is the very basis of data encryption used by the public on a daily basis via the World Wide Web. My proposed project aims to understand taut foliations and other related notions, and to continue the work of Agol, Hass and Thurston for furthering our understanding of the knot genus questions. This project will create new bridges between different areas of mathematics and computer science and can potentially have important applications to the study of DNA, and our understanding of the P vs NP question.