Poly-Time Knot Theory and Quantum Algebra

其中一个重大的胜利数学在20世纪80年代，导致至少3菲尔兹奖章（琼斯，Drinfel'd，维滕）是意想不到的实现，即低维拓扑，特别是结理论，是密切相关的量子场论和理论的量子群。纽结理论是世俗的，古老的;任何“量子”似乎超现代。这两个人怎么会有关系？这个问题的答案很长，很复杂，与“杨-巴克斯特方程”（YBE）有很大关系。一方面，YBE在纽结理论中可以被解释为“第三个Reidemeister移动”，或者“控制3根弦的最基本的相互作用”（这是纽结理论中非常关键的部分）。另一方面，YBE的解决方案来自“量子”机制。因此，量子对打结拓扑有用，对低维拓扑的其余部分也是如此，但“量子”有一个警告，这使得它（对某些人）超级令人兴奋，但（对另一些人）限制了它的有用性。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，可能需要很多年的时间，它们所能做的事情受到其他限制，而且大部分低维拓扑都在这些限制之外。因此，至少从现在开始，而且很可能永远如此，许多描述中包含“量子”的东西，计算起来都是指数复杂的，这在实践中意味着，除了几个简单的例子，它们无法计算。最近，货车德维恩和我，继罗赞斯基和奥弗贝之后，发现了一个角落（形象地说）在纽结理论中使用的量子机械的巨大状态空间，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支持纽结理论。这样构造的“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计算的。我们的方法利用了复杂的对称群通常具有更简单的“收缩”的事实。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，杨-巴克斯特方程的有用解的基础上的对称代数，即半单代数，如sl（n），具有“可解代数”的压缩，并且对于原始sl（n）对称性是指数复杂的相同运算变成多项式复杂的（即，简单得多）在这些可解压缩之内和附近。还有很多工作要做：实现，文档，应用，推广。我希望在这五年的资助期内实现这一切。

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.