Quantum algebra and topology

量子代数和拓扑

• 批准号: 0072342
• 负责人: Greg Kuperberg
• 金额: $ 7.11万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072342&HistoricalAwards=false
• 关键词: Quantum; algebra; topology

Proposal: DMS-0072342PI: Greg KuperbergThe purpose of the project is to study the properties and applications ofquantum groups, the Yang-Baxter equation, graph homology, and similarconstructions in algebra with geometric formalism. In particular the role ofthese constructions will be considered in two areas: In 3-dimensionaltopology, they arise in the guise of of perturbative and non-perturbativeinvariants of Chern-Simons field theory. There is a good chance that suchinvariants are related to the Thurston Geometrization Program, at least forhyperbolic manifolds. In quantum computation, the same non-perturbativeChern-Simons field theory produces an interesting method for fault tolerance,one which needs to be developed further.The purpose of the project is to develop and apply certain areas of algebra inwhich formulas have a geometric structure. Two famous examples of this are alinear algebra, in which matrices have some geometry, and Feynman diagrams,whose geometry is physically meaningful. Geometrization of algebraic formulashas been influential in enumerative combinatorics, in non-commutative algebra,and in geometric topology in the last 15 years. One of the first and mostimportant representatives of this recent trend is the Jones polynomial, aninvariant of knots. Although the Jones polynomial arose in algebra andmathematical, it and its generalization are at least as important in topology,and recently also in quantum computation. These applications of the new trendin algebra will be investigated.

提案：DMS-0072342PI：Greg Kuperberg 该项目的目的是研究量子群、Yang-Baxter 方程、图同调以及几何形式主义代数中的类似结构的性质和应用。 特别是这些结构的作用将在两个领域被考虑：在3维拓扑中，它们以陈-西蒙斯场论的微扰和非微扰不变量的形式出现。 这些不变量很可能与瑟斯顿几何化程序有关，至少对于双曲流形来说是这样。 在量子计算中，同样的非微扰陈-西蒙斯场论产生了一种有趣的容错方法，该方法需要进一步开发。该项目的目的是开发和应用代数的某些领域，其中公式具有几何结构。 两个著名的例子是线性代数和费曼图，线性代数中的矩阵具有一定的几何意义，费曼图的几何意义具有物理意义。在过去的 15 年里，代数公式的几何化对枚举组合学、非交换代数和几何拓扑产生了影响。 这一最新趋势的第一个也是最重要的代表之一是琼斯多项式，它是纽结的不变量。 尽管琼斯多项式出现在代数和数学中，但它及其推广至少在拓扑学中同样重要，最近在量子计算中也同样重要。 将研究新趋势代数的这些应用。