The Topology of Quantum Invariants

DMS-0103922贾斯汀·D·罗伯茨这个项目的目标是试图理解三维量子不变量背后的拓扑。目前的知识状态似乎不足以解释不变量携带什么样的拓扑信息，因此它们在三维拓扑中可能有什么样的应用。然而，它揭示了(通过TQFT和Vassiliev理论的框架)丰富的非常有趣的代数结构，这些结构似乎特定地与三维拓扑有关，因此应该被认为是一种三维“新代数拓扑”。Roberts打算试图弥合这种新拓扑与经典代数拓扑之间的差距。有三个主要的调查领域。第一个是K理论，它可以帮助解释量子纽结不变量和瓦西里耶夫不变量之间的关系(例如，为什么琼斯多项式是多项式)。第二个是Rozansky-Witten理论，这是一个非常有趣的TQFT的新例子，它提供了关于不变量解释的新线索，并可能在复杂或Hyperkaehler几何中有更多的应用。第三种是Kashaev、Murakami和Murakami的双曲体积猜想。这一猜想似乎符合将量子化(特别是6J符号)与经典几何联系起来的一般概念模式；这些概念可能为该猜想提供概念背景。对数学家来说，结是你将一根绳子缠绕在一起，然后将其末端粘合在一起得到的东西。很明显，有不同的方法来做这件事；试图描述和理解这些方法形成了拓扑学的一个分支，当定量问题被忽略时，拓扑学可能被认为是几何学的残余。纽结理论的主要工具是“拓扑不变量”，即可以与纽结联系在一起的数字，它编码了关于纽结结构的一些信息。在过去的16年里，从量子场论物理学中借用的思想导致了许多新的和美丽的不变量被称为量子不变量，但由于这个不同寻常的起源，它们的意义仍然非常神秘，它们的潜力没有得到充分发挥。该项目的目标是使用一些新的想法来试图解释这个谜团，并弥合几何、拓扑和物理之间的一些差距。这项工作应该有助于这些学科之间健康的思想交流，并在每个学科中激发新的工作。

DMS-0103922Justin D. Roberts The goal of this project is to try to understand the topology underlying quantum invariants in three dimensions. The current state of knowledge seems inadequate for explaining what kind of topological information the invariants carry, and therefore what kind of applications in three-dimensional topology they might have. However ithas revealed (through the frameworks of TQFT and Vassiliev theory) a wealth of very interesting algebraic structures, which seem to be specifically related to topology in three dimensions, and so should be thought of as kind of ``new algebraic topology'' in three dimensions.Roberts intends to try to bridge the gap between this new topology and classical algebraic topology. There are three main areas of investigation. The first is K-theory, which could help to explain the relationship between quantum knot invariants and Vassiliev invariants (for example, why the Jones polynomial is a polynomial). The second is Rozansky-Witten theory, a very interesting new example of a TQFT which provides new hints about intepretation of the invariants, and could have additional applications in complex or hyperkaehler geometry. The third is the hyperbolic volume conjecture of Kashaev, Murakami and Murakami. This conjecture seems to fit into a general pattern of ideas relating quantization (and specifically, 6j-symbols) to classicalgeometry; ideas which might provide a conceptual context for the conjecture.To a mathematician, a knot is what you get by tangling up a piece of string and then gluing its ends together. It is clear that there are qualitatively different ways of doing this; trying to describe and understand these ways forms a branch of topology, which might be thought of as what remains of geometry when quantitative questions are ignored. The main tools of knot theory are "topological invariants", numbers which can be associated to knots and which encode someinformation about their structure. During the last sixteen years, ideas borrowed from the physics of quantum field theory have led to the discovery of many new and beautiful invariants called "quantum invariants", but because of this unusual origin, their meaning remains very mysterious, and their potential unfulfilled. The goal of the project is to use some new ideas to try to explain this mystery and tobridge some of the gaps between geometry, topology, and physics. The work should contribute to the healthy exchange of ideas between these disciplines, and stimulate new work in each.