Algorithmic topology in low dimensions

低维算法拓扑

• 批准号: EP/Y004256/1
• 负责人: Marc Lackenby
• 金额: $ 193.11万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY004256%2F1
• 关键词: Algorithmic; topology; low; dimensions

Low-dimensional topology is a hugely active and influential area of modern mathematical research. Knots, which are just simple closed curves embedded in 3-dimensional space, play a central role in the subject. Two knots are 'equivalent' if one can be deformed into the other without the curve passing through itself. The way that knots are usually specified is by means of a 2-dimensional 'diagram' which encodes a projection of the knot to a plane. A basic question in the field is: given two knot diagrams, can we reliably decide whether the knots are equivalent? In effect, we are asking for an algorithm to solve this problem. This is one of the primary questions in the field of algorithmic topology, which is the main focus of this research proposal. This problem is known to be solvable, but the fastest known algorithm has incredibly huge running time: it is a tower of exponentials, with some fixed but unknown height. One of the main goals of the project is to provide a dramatic improvement to this. It is possible that there is a universal polynomial p, with the property that the two knot diagrams with n and m crossings are related by p(n) + p(m) Reidemeister moves. These moves are simple modifications to the diagram that do not change the knot type. If so, this would provide an exponential-time algorithm for the equivalence problem, and would establish that it lies in the complexity class NP (Non-deterministic Polynomial time). Problems in NP are those for which a positive answer can be easily demonstrated.A major theme in low-dimensional topology is the use of knot 'invariants', which are mathematical quantities (such as polynomials) that can be assigned to a knot. They have the property that if two knots are equivalent, then they have the same invariants. There are now countless different knot invariants, that are defined using very diverse areas of mathematics, such as quantum field theory or non-Euclidean geometry. In a recent breakthrough, the PI and his collaborators have used techniques from the field of Artificial Intelligence to discover new connections between these invariants. One of the main goals of the fellowship is to develop these techniques, to find new connections. This is a methodology that is undoubtedly very general, and that will have applications to many different branches of mathematics.

低维拓扑是现代数学研究中一个非常活跃和有影响力的领域。节点是嵌入在三维空间中的简单闭合曲线，在主题中扮演着核心角色。如果两个结可以变形成另一个，而曲线不通过它自己，那么两个结是“等价的”。节点通常是通过二维“图”来指定的，该图编码了节点到平面的投影。这个领域的一个基本问题是：给出两个结图，我们能可靠地确定这两个结是否等价吗？实际上，我们正在要求一种算法来解决这个问题。这是算法拓扑学领域的主要问题之一，也是本研究方案的主要关注点。这个问题众所周知是可以解决的，但已知的最快算法的运行时间令人难以置信地巨大：它是一个指数塔，有一些固定但未知的高度。该项目的主要目标之一是为这一点提供戏剧性的改善。有可能存在一个普适多项式p，它的性质是具有n和m个交叉的两个纽结图通过p(N)+p(M)Reidemister移动相关。这些移动是对关系图的简单修改，不会更改节点类型。如果是这样的话，这将为等价问题提供一个指数时间算法，并将确定它属于复杂性类NP(非确定性多项式时间)。NP中的问题是那些可以很容易地证明肯定答案的问题。低维拓扑学的一个主要主题是使用节点不变式，这是可以赋给节点处的数学量(如多项式)。它们具有这样的性质：如果两个纽结等价，则它们具有相同的不变量。现在有无数不同的纽结不变量，它们是用非常不同的数学领域定义的，比如量子场论或非欧几里德几何。在最近的一项突破中，PI和他的合作者使用了人工智能领域的技术来发现这些不变量之间的新联系。该奖学金的主要目标之一是开发这些技术，找到新的联系。这无疑是一种非常普遍的方法论，它将适用于数学的许多不同分支。