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。结只是嵌入 3 维空间中的简单闭合曲线，在该主题中发挥着核心作用。如果一个结可以变形为另一个结而曲线不穿过自身，则两个结是“等效的”。通常指定结的方式是通过二维“图”来编码结到平面的投影。该领域的一个基本问题是：给定两个结图，我们能否可靠地确定这两个结是否等价？实际上，我们正在寻求一种算法来解决这个问题。这是算法拓扑领域的主要问题之一，也是本研究提案的主要焦点。众所周知，这个问题是可以解决的，但已知最快的算法具有令人难以置信的巨大运行时间：它是一座指数塔，具有固定但未知的高度。该项目的主要目标之一是对此进行显着改进。可能存在一个通用多项式 p，其性质是具有 n 和 m 个交叉点的两个结图通过 p(n) + p(m) Reidemeister 移动相关。这些动作是对图表的简单修改，不会改变结类型。如果是这样，这将为等价问题提供指数时间算法，并确定它属于复杂性类别 NP（非确定性多项式时间）。 NP 中的问题是那些可以轻松证明肯定答案的问题。低维拓扑中的一个主要主题是使用结“不变量”，它们是可以分配给结的数学量（例如多项式）。它们具有这样的性质：如果两个结相等，则它们具有相同的不变量。现在有无数不同的结不变量，它们是使用非常不同的数学领域来定义的，例如量子场论或非欧几里得几何。在最近的一项突破中，PI 和他的合作者使用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，寻找新的联系。毫无疑问，这是一种非常通用的方法，并且可以应用于许多不同的数学分支。