Algorithmic topology in low dimensions

低维拓扑学是现代数学研究中一个非常活跃和有影响力的领域。结是嵌入三维空间的简单闭合曲线，在这一主题中起着核心作用。两个结点是“等效的”，如果其中一个可以变形成另一个，而曲线不会穿过它自己。结点通常是通过二维“图”来指定的，它将结点的投影编码到平面上。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结是否等效？实际上，我们是在要求一个算法来解决这个问题。这是算法拓扑领域的主要问题之一，也是本课题研究的重点。众所周知，这个问题是可以解决的，但已知最快的算法运行时间非常长：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就是对这一点进行显著改进。有可能存在一个泛多项式p，其性质是有n和m个交叉点的两个结图通过p(n) + p(m) Reidemeister移动相关联。这些动作是对图的简单修改，不会改变结的类型。如果是这样，这将为等价问题提供一个指数时间算法，并将其建立在复杂度类NP（非确定性多项式时间）中。NP中的问题是那些可以很容易证明其正答案的问题。低维拓扑中的一个主要主题是使用结“不变量”，这是可以分配给结的数学量（如多项式）。它们有这样的性质如果两个结点是等价的，那么它们就有相同的不变量。现在有无数不同的结不变量，它们是用非常不同的数学领域来定义的，比如量子场论或非欧几里得几何。在最近的一项突破中，PI和他的合作者利用人工智能领域的技术发现了这些不变量之间的新联系。这项研究的主要目标之一就是发展这些技术，寻找新的联系。毫无疑问，这是一种非常普遍的方法，它将应用于许多不同的数学分支。

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.