Non-commutative algebra and relations to topology and quantum information theory

主要研究人员：格雷戈里·库珀伯格建议编号：0306681机构：加州大学戴维斯分校标题：非交换代数及其与拓扑学和量子信息理论的关系摘要：本项目的目标是发展非交换(或量子)代数的概念，因为它们应用于低维拓扑和量子信息理论。PI计划研究张量范畴，特别是融合范畴和来自量子群的范畴，这些范畴可以用来构造3-流形不变量。PI还计划研究微扰量子不变量(由于Vassiliev和Kontsevich)和张量范畴形变理论的相关问题。在量子信息论领域，PI计划研究概率论中基本结果的类似，如中心极限定理，以及表象理论和量子群的可能应用。现代数学和物理的基础之一是非对易代数，它研究乘法不交换的抽象公式(尽管加法仍然可以交换)。非对易代数是量子力学的本质，也是理解对称性的基础，本项目的目的是进一步将非对易代数应用到量子计算和三维拓扑中。量子计算既是量子力学的重要理论阐述，也是一种可能的未来技术。拓扑学是研究节点、链接和抽象几何空间的学科。它与对称性密切相关，它还应用于数学的许多其他领域，以及物理学和经济学中的问题，甚至DNA的研究。

Principal Investigator: Gregory KuperbergProposal Number: 0306681Institution: University of California-DavisTitle: Non-commutative algebra and relations to topology and quantum information theoryABSTRACT: The goal of this project is to develop concepts in non-commutative (or quantum) algebra as they apply to low-dimensional topology and quantum information theory. The PI plans to study tensor categories, in particular fusion categories and categories arising from quantum groups, that can be used to construct 3-manifold invariants. The PI also plans to study perturbative quantum invariants (due to Vassiliev and Kontsevich) and the related problem of deformation theory of tensor categories. In the area of quantum information theory, the PI plans to study analogues of foundational results in probability theory such as the central limit theorem, as well as possible applications of representationtheory and quantum groups.One of the foundations of modern mathematics and physics is non-commutative algebra, which is the study of abstract formulas in which multiplication does not commute (although addition still does). Non-commutative algebra is the essence of quantum mechanics, and it is also fundamental for understanding symmetry, The purpose of this project is to further apply non-commutative algebra to quantum computation and 3-dimensional topology. Quantum computation is at once an important theoretical elaboration of quantum mechanics and a possible future technology. Topology is the study of knots, links, and abstract geometric spaces. It is closely related to symmetry and it has applications to many other areas of mathematics, as well as to questions in physics and economics and even to the study of DNA.