RUI: Quantum Invariants of Three-Manifolds and Their Applications

提议：DMS-9971905首席研究员：Joanna Kania-Bartoszynska摘要：量子拓扑学的一个基本问题是为三维流形的新不变量找到拓扑解释。这个项目的第一部分将量子不变量放在纽结理论和表象理论的背景下。在此背景下，作者打算用基本群的表示来分析三维流形的量子不变量的渐近行为。第二部分涉及应用量子不变量来形成3-流形中对称性存在的准则。量子拓扑学的领域始于十多年前，数学家V·琼斯发现了琼斯多项式，物理学家E·威腾用拓扑量子场论解释了这一多项式。今天，它仍然是一个充满活力的研究领域，它汇集了物理、代数、几何和拓扑学的概念和技术。这个项目的目的是进一步调查这些想法是如何结合在一起的。

Proposal: DMS-9971905 Principal Investigator: Joanna Kania-Bartoszynska Abstract:A fundamental problem of quantum topology is that of finding topological interpretations for the new invariants of 3-manifolds. The first part of this project places quantum invariants in the context of knot theory and representation theory. The investigator intends to use this context to analyze the asymptotic behavior of the quantum invariants of a 3-manifold in terms of representations of its fundamental group. The second part involves applying the quantum invariants to form criteria for the existence of symmetries in a 3-manifold. The field of quantum topology began just over a decade ago with the discoveries of the Jones polynomial, by the mathematician V. Jones, and with the physicist's E. Witten's explanation of this polynomial in terms of the topological quantum field theory. Today it continues to be a dynamic area of research bringing together constructs and techniques from physics, algebra, geometry, and topology. The purpose of this project is to investigate further how these ideas fit together.