Poly-Time Knot Theory and Quantum Algebra

多时间结理论和量子代数

• 批准号: RGPIN-2018-04350
• 负责人: BarNatan, Dror
• 金额: $ 2.04万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=682697
• 关键词: Poly; Time; Knot; Theory; Quantum

One of the major triumphs of mathematics in the 1980s, which lead to at least 3 Fields medals (Jones, Drinfel'd, Witten) was the unexpected realization that low dimensional topology, and in particular knot theory, is closely related to quantum field theory and to the theory of quantum groups. Knot theory is mundane and ages-old; anything "quantum" seems hyper-modern. Why would the two have anything to do with each other?******The answer is long and complicated and has a lot to do with the "Yang-Baxter Equation" (YBE). The YBE on the one hand can be interpreted in knot theory as "the third Reidemeister move", or as "controlling the most basic interaction of 3 pieces of string" (this turns out to be a very crucial part of knot theory). On the other hand solutions of the YBE arise from "quantum" machinery. Hence the quantum is useful to the knotted, and by similar ways, to the rest of low dimensional topology.******But "quantum" has a caveat, which makes it super-exciting (to some) yet bounds its usefulness (to others). When quantum systems grow large (as they do when the knot or low-dimensional space we study grows complicated), their "state space" grows at an exponential rate. "Quantum computers" aim to exploit this fact and make large quantum systems performs overwhelmingly large computations by utilizing their vast state spaces. But quantum computers aren't here yet, may take many years to come, suffer from other limits on what they can do, and much of low-dimensional topology is anyway outside of these limits. So at least for now and likely forever, many things that have "quantum" in their description are exponentially-complex to compute, which in practice means that they cannot be computed beyond a few simple cases.******Recently Van der Veen and myself, following Rozansky and Overbay, found a corner (figuratively speaking) of the vast state space of the quantum machinery used in knot theory, which can be described in just polynomial complexity, and which carries enough information to still speak to knot theory. The "knot invariants" constructed that way seem to be the strongest invariants we know that are computable even for very large knots.******Our approach utilizes the fact that complicated symmetry groups often have much simpler "contractions". A well known example is the Lorentz group of relativity theory, which at small velocities contracts to the Galilean group of classical mechanics. In a similar manner we find that the symmetry algebras underlying the useful solutions of the Yang-Baxter equation, namely semi-simple algebras such as sl(n), have contractions that are "solvable algebras", and that the same operations that are exponentially complex for the original sl(n) symmetry become polynomially-complex (namely, much simpler) within and near these solvable contractions.******Much remains to be done: implementation, documentation, application, generalization. I hope to achieve all that over this 5-year grant period.

20世纪80年代数学的主要成就之一是意外地认识到低维拓扑，特别是纽结理论，与量子场论和量子群理论密切相关，这导致了至少三个菲尔兹奖(琼斯、德伦费尔、威腾)。纽结理论是平凡而古老的；任何“量子”似乎都是超现代的。为什么这两者会有任何关系呢？答案既长又复杂，与“杨-巴克斯特方程”(YBE)有很大关系。在纽结理论中，纽结一方面可以被解释为“第三步莱德迈斯特动作”，或者“控制三根弦的最基本的相互作用”(这被证明是纽结理论中非常关键的部分)。另一方面，“可能”的解决方案来自“量子”机制。因此，量子对于结点和低维拓扑的其余部分都是有用的。*但是“量子”有一个警告，这使得它非常令人兴奋(对一些人)，但也限制了它的有用性(对另一些人)。当量子系统变大时(就像我们研究的结点或低维空间变得复杂时)，它们的“状态空间”会以指数速度增长。“量子计算机”旨在利用这一事实，使大型量子系统通过利用其巨大的状态空间来执行压倒性的大规模计算。但量子计算机还没有出现，可能还需要很多年的时间，它们的功能还受到其他限制，而且许多低维拓扑结构无论如何都超出了这些限制。因此，至少就目前而言，也可能永远如此，许多描述中含有“量子”的事物对于计算来说都是指数复杂的，这实际上意味着它们不能超越几种简单的情况进行计算。*最近，范德维恩和我跟随罗赞斯基和奥弗贝，发现了纽结理论中所使用的量子力学的巨大状态空间的一个角落(比喻地说)，它可以用多项式的复杂性来描述，并且携带了足够的信息，仍然可以谈论纽结理论。以这种方式构造的“纽结不变量”似乎是我们所知的最强的不变量，即使对于非常大的纽结也是可以计算的。*我们的方法利用了这样一个事实，即复杂的对称群通常有更简单的“收缩”。一个著名的例子是相对论的洛伦兹群，它以较小的速度收缩到伽利略经典力学群。以类似的方式，我们发现作为Yang-Baxter方程有用解的基础的对称代数，即半单代数，如sl(N)，具有“可解代数”的压缩，并且在这些可解压缩的内部和附近，对于原始sl(N)对称是指数复数的相同运算变成多项式复数(即，简单得多)。*仍有许多工作要做：实现、文献、应用、推广。我希望在这5年的资助期内实现所有这些目标。