Mathematical Sciences: Holomorphic Invariants of 3-Manifolds

小行星9626544 三维流形的Witten-Reshetikhin-Turaev（WRT）量子不变量族被定义为依赖于一个参数的离散不变量族，该参数可以是任何单位根。 本项目旨在研究这个不变量到参数的全纯函数的推广。 根据村上，Ohtsuki，Rozansky和首席研究员的工作，目前可以定义这样的扩展仅适用于特定的流形。 该项目的目标是更好地理解这个函数之间的关系，它采取的特定值，它的渐近展开，以及原始流形的几何和拓扑的组合描述。 这将导致有限型3流形不变量的组合公式，广义的卡森不变量，以及离散化的陈-西蒙斯-维滕费曼积分和比较与Kontsevich积分。 虽然我们对所谓的三维纽结和链环的量子不变量家族（通常是复参数的多项式函数）有很多了解，但目前对它们的三维流形对应物还没有同样的了解。 从少数已知的结果，这些功能，他们被认为有有趣的数论性质，调查这是本项目的主题。 这些不变量也有一个公式作为费曼积分，希望这个项目将有助于更好地理解费曼积分一般，目前坐在一个不完全严格的基础。 ***

9626544 Lawrence The Witten-Reshetikhin-Turaev (WRT) family of quantum invariants of 3-manifolds was defined only as a family of discrete invariants dependent on a parameter that can be any root of unity. This project aims to study the extension of this invariant to a holomorphic function of the parameter. Following work of Murakami, Ohtsuki, Rozansky and the Principal Investigator, it is currently possible to define such an extension only for specific manifolds. The goal of the project is to understand better the relationship between this function, specific values that it takes, its asymptotic expansion, and a combinatorial description of the geometry and topology of the original manifold. This should lead to combinatorial formulae for finite type 3-manifold invariants, generalizations of the Casson invariant, as well as a discretization of the Chern-Simons-Witten Feynman integral and comparisons with the Kontsevich integrals. Although much is known about the so-called quantum family of invariants of knots and links in three dimensions, which are generally polynomial functions of a complex parameter, the same is currently not true for their 3-manifold counterparts. From the few known results on these functions, they are seen to have intriguing number theoretic properties, the investigation of which is the subject of this project. These invariants also have a formulation as a Feynman integral, and it is hoped that this project will contribute to the better understanding of Feynman integrals in general, which currently sit on a not entirely rigorous foundation. ***