Algorithmic topology in low dimensions

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。纽结，这只是简单的封闭曲线嵌入在三维空间中，发挥了核心作用的主题。如果一个节点可以变形为另一个节点，而曲线不穿过它本身，那么两个节点是“等价的”。通常指定结的方式是通过二维“图”，该图编码结到平面的投影。该领域的一个基本问题是：给定两个纽结图，我们能否可靠地判定这两个纽结是否等价？实际上，我们正在寻求一种算法来解决这个问题。这是算法拓扑学领域的主要问题之一，也是本研究提案的主要焦点。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提供一个戏剧性的改善。有可能存在一个泛多项式p，其性质是两个具有n和m个交叉点的纽结图通过p（n）+ p（m）Reidemeister移动而相关。这些移动是对图的简单修改，不会更改结类型。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）。它们具有这样的性质：如果两个纽结是等价的，那么它们具有相同的不变量。现在有无数不同的纽结不变量，它们是使用非常不同的数学领域定义的，例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发展这些技术，找到新的联系。这无疑是一种非常普遍的方法论，它将应用于数学的许多不同分支。

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.