Quantum Mathematics and Geometry

这项拟议的研究将探索量子计算和量子信息理论的数学思想，以及数学本身的相关思想。量子理论数学的一个重要特征是非对易，“A乘以B”的概念并不总是等于“B乘以A”。这反映在量子力学的实验中，测量的结果取决于它们被测量的顺序；此外，量子力学中的测量本质上是统计的。非对易现象导致了令人惊讶的新算法，如果有一天它们被构建起来，这些算法可以在量子计算机上运行。在数学本身，量子式的非对易性在几何学和拓扑学中产生了许多新的概念，例如新的对称性概念，例如描述和区分纽结和类似拓扑对象的新方法。PI计划调查所有这些类型的量子数学之间的联系和新的结构。

The proposed research will explore the mathematical ideas of quantum computation and quantum information theory, and related ideas in mathematics itself. A defining feature of the mathematics of quantum theory is non-commutativity, the idea the "A times B" does not always equal "B times A". This is reflected in experiments in quantum mechanics that the outcomes of measurements depends on the order in which they are measured; moreover, measurements in quantum mechanics are inherently statistical. The phenomenon of non-commutativity leads to surprising new algorithms that could run on quantum computers if they are one day constructed. In mathematics itself, quantum-style non-commutativity has led to many new ideas in geometry, such as new notions of symmetry, and in topology, such as new methods to describe and distinguish knots and similar topological objects. The PI plans to investigate connections and new constructions among all of these types of quantum mathematics.