RUI: Quantum Invariants of Three-Manifolds and Their Applications

提案：DMS-9971905 主要研究者：Joanna Kania-Bartoszynska 翻译后摘要：量子拓扑学的一个基本问题是，找到新的3-流形的不变量的拓扑解释。 这个项目的第一部分将量子不变量置于纽结理论和表象理论的背景下。研究者打算利用这一背景来分析三维流形的量子不变量在其基本群表示方面的渐近行为。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。 量子拓扑学的研究始于十多年前，当时数学家V. Jones发现了琼斯多项式，物理学家E。维滕用拓扑量子场论对这个多项式的解释。 今天，它仍然是一个充满活力的研究领域，汇集了来自物理学，代数学，几何学和拓扑学的结构和技术。 这个项目的目的是进一步研究这些想法如何结合在一起。

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.