Algorithmic topology in low dimensions

点击翻译按钮获取中文摘要

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.