Poly-Time Knot Theory and Quantum Algebra

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，特别是结理论，与量子场论和量子群理论密切相关，这至少导致了3个菲尔兹奖（琼斯，德林费尔德，威滕）。绳结理论是平凡而古老的；任何“量子”似乎都是超现代的。这两个人怎么会有什么关系呢？******答案很长很复杂，与“杨-巴克斯特方程”（YBE）有很大关系。一方面，YBE可以在结理论中解释为“第三个Reidemeister移动”，或者“控制3根弦的最基本的相互作用”（这被证明是结理论的一个非常关键的部分）。另一方面，YBE的解来自“量子”机制。因此，量子对于打结的结构是有用的，并且以类似的方式，对于其他低维拓扑结构也是有用的。******但是“量子”有一个警告，这使得它（对一些人来说）超级令人兴奋，但也限制了它的实用性（对另一些人来说）。当量子系统变大时（就像我们研究的结或低维空间变得复杂时一样），它们的“状态空间”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型计算。但是量子计算机还没有出现，可能还需要许多年的时间，在它们能做的事情上受到其他限制，而且许多低维拓扑无论如何都超出了这些限制。因此，至少在现在，而且很可能永远，许多在描述中带有“量子”的事物都是指数级复杂的，这意味着在实践中，它们无法计算出几个简单的情况。******最近，Van der Veen和我在Rozansky和Overbay之后，发现了用于打结理论的量子力学的巨大状态空间的一个角落（形象地说），它可以用多项式复杂度来描述，并且它携带了足够的信息来谈论打结理论。以这种方式构造的“结不变量”似乎是我们所知道的最强的不变量，即使对于非常大的结也是可以计算的。******我们的方法利用了一个事实，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，它在小速度下收缩到经典力学的伽利略群。以类似的方式，我们发现Yang-Baxter方程有用解的对称代数，即半简单代数，如sl(n)，具有“可解代数”的收缩，并且对于原始sl(n)对称来说是指数复的相同操作在这些可解的收缩内和附近变得多项式复（即更简单）。******还有很多工作要做：实现、文档、应用、推广。我希望在这5年的资助期内，能达到所有这些目标。

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.