Quantum Methods in Mathematics and Computation

这个项目的目的是探索量子方法，或非对易代数和非对易概率的方法，在传统的非量子环境中。特别是，该项目将探索不变张量的特殊基，作为量子三维流形不变量极限的双曲结构，量子算法，以及量子度量空间的概念。相关的衍生项目包括使用中心极限定理对概率中的概率方法进行扩展，以及对复杂类(经典和量子)之间的已知关系进行分类。该项目的目的是将量子物理的思想应用于数学，特别是几何和拓扑学，以及计算机科学。量子力学的核心是量子不确定性的统计概念。这是一个想法和事实，当一个物体的一个物理测量(如位置)是精确知道的，其他相关的值(如动量)不能有确定值，而是有一个统计范围的值。量子不确定性总是由非对易的数学概念来建模，即当A和B是抽象变量时，AB不需要等于BA。非交换性在低维几何和拓扑学中发挥着越来越重要的作用。量子不确定性还可能导致能够执行特殊量子算法的新型计算机。这两个研究趋势在智力上是相似的，这个项目将一起探索它们。

The aim of this project is to explore quantum methods, or methods of non-commutative algebra and non-commutative probability, in traditionally non-quantum settings. In particular, the project will explore special bases of invariant tensors, hyperbolic structure as a limit of quantum 3-manifold invariants, quantum algorithms, and notions of quantum metric spaces. Related spin-off projects include an extension of the probabilistic method in probability using the central limit theorem, and a classification of known relations among complexity classes (both classical and quantum).The aim of this project is to apply the ideas of quantum physics to both mathematics, in particular to geometry and topology, and to computer science. The heart of quantum mechanics is the statistical notion of quantum uncertainty. This is the idea and the fact that when one physical measurement of an object (such as position) is known precisely, other related values (such as momentum) cannot have definite values, but instead have a statistical range of values. Quantum uncertainty is always modelled by the mathematical notion of non-commutativity, which is the idea that AB need not equal BA when A and B are abstract variables. Non-commutativity plays an increasingly important role in low-dimensional geometry and topology. Quantum uncertainty could also lead to new kinds of computers that can execute special quantum algorithms. These two currents of research are intellectually similar, and this project will explore them together.