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

结属的拉紧叶状、表示和计算复杂性

基本信息

批准号：
EP/T016582/1
负责人：
Mohammadmahdi Yazdi
金额：
$ 37.53万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT016582%2F1
关键词：
Taut foliations representations computational complexity

项目摘要

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.

该项目涉及拓扑学（形状的数学研究）和计算复杂性（如何有效地使用计算机解决问题）领域。这个项目从三流形的研究开始。“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，想象一个封闭的，可能纠结的绳子（例如，一个结）在3D空间中的补。如果我们可以不断地将一个结变形为另一个结而不撕裂第一个结，我们（拓拓学家）认为这两个结是同一个结。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，我们也看二维流形，被称为“曲面”，局部看起来像几何平面。球体和甜甜圈（即有一个手柄的环面）是表面最简单的例子。如果我们从每个表面上取下一个小圆盘，我们就会得到我们所说的“有边界的表面”。这个曲面的“属”是它拥有的手柄的数量。拓扑学家已经知道，结的一个重要的拓扑特征确实是“结属”。我们可以将结的属(K)定义为边界与K重合的所有可能纠结的可定向表面之间的最小属。确定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。Agol， Hass和Thurston表明，如果我们允许结和周围的三流形变化，那么“结属”的问题是“np完全的”。“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，在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，如果我们找到了其中一个问题的解决方案，那么我们就拥有了解锁所有问题的万能钥匙！因此，一个np完全问题的解决方案是通往跨越计算知识的大量重要答案的门户。这是数学最迷人之处之一：我们的工作可能将这些遥远的现象统一起来。到目前为止，确定结属的唯一实用方法涉及所谓的“叶理理论”。该理论的术语灵感来自地理学中的分层岩石，这给我们谈论的这种空间的永恒性质和无限提供了一个很好的视觉效果。接下来，我们将三流形的叶面化理解为将三流形分割成表面（称为“叶子”，这个术语的灵感来自于树叶），这样在局部，这些表面就像一叠纸一样合在一起。这里需要注意的是，曲面也可以是无限的，我们在这里不讨论。有一类特别重要的叶理叫做“紧叶理”。直观地说，紧绷的叶子具有这样的特性，即所有叶子的面积都最小化（就像“肥皂膜”，当两个肥皂泡合并并在它们之间形成薄膜时形成）。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与NP是克莱研究所颁发的千禧年百万美元大奖之一，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概念，并继续Agol， Hass和Thurston的工作，以进一步了解结属问题。这个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的研究和我们对P / NP问题的理解有重要的应用。