Quantum algebra and topology

提案：DMS-0072342PI: Greg kuperberg项目的目的是研究量子群、Yang-Baxter方程、图同调和几何形式代数中的相似结构的性质和应用。特别是这些结构的作用将在两个方面加以考虑：在三维拓扑中，它们以chen - simons场论的微扰不变量和非微扰不变量的形式出现。有一个很好的机会，这样的变体是有关瑟斯顿几何化程序，至少对于双曲流形。在量子计算中，同样的非摄动场理论产生了一种有趣的容错方法，这种方法需要进一步发展。该项目的目的是发展和应用代数的某些领域，其中公式具有几何结构。这方面有两个著名的例子，一个是线性代数，其中矩阵有一些几何形状，另一个是费曼图，其几何形状在物理上是有意义的。在过去的15年中，代数公式的几何化在枚举组合学、非交换代数和几何拓扑学中产生了很大的影响。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，它是结的不变量。虽然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。我们将研究代数新趋势的这些应用。

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.