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

非交换代数以及与拓扑和量子信息论的关系

• 批准号: 0306681
• 负责人: Greg Kuperberg
• 金额: $ 10万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306681&HistoricalAwards=false
• 关键词: Non; commutative; algebra; relations; topology

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.

首席研究员：Gregory Kuperberg 提案编号：0306681 机构：加州大学戴维斯分校 标题：非交换代数及其与拓扑和量子信息论的关系 摘要：该项目的目标是发展非交换（或量子）代数中的概念，因为它们适用于低维拓扑和量子信息论。 PI 计划研究张量类别，特别是融合类别和由量子群产生的类别，可用于构造 3 流形不变量。 PI 还计划研究微扰量子不变量（由 Vassiliev 和 Kontsevich 提出）以及张量类别变形理论的相关问题。 在量子信息论领域，PI计划研究概率论基础结果的类似物，例如中心极限定理，以及表示论和量子群的可能应用。现代数学和物理学的基础之一是非-交换代数，是对抽象公式的研究，其中乘法不能交换（尽管加法仍然可以交换）。非交换代数是量子力学的本质，也是理解对称性的基础，本项目的目的是将非交换代数进一步应用到量子计算和3维拓扑中。 量子计算既是量子力学的重要理论阐述，也是未来可能的技术。 拓扑学是对结、链接和抽象几何空间的研究。 它与对称性密切相关，并且可应用于许多其他数学领域，以及物理和经济学问题，甚至 DNA 研究。