Asymptotics of Quantum Invariants

该项目是三维拓扑和几何一般领域的一部分，该领域分析具有高度复杂性的三维空间，并包括对三维曲线发生的打结现象的研究。潜在的问题经常出现在我们生活的三维世界的实际应用中，但也来自数学物理中更多的理论考虑。该项目调查了两种截然不同的三维拓扑学方法之间令人惊讶的联系。第一种方法是非常代数的，涉及到三维空间中纽结曲线的所谓量子不变量。第二种方法更具几何性和解析性，并且使用了三维双曲空间的非欧几里德几何。卡沙耶夫体积猜想在许多例子中得到了实验验证，但仍然没有数学证明保证它在所有情况下都成立，它在这两个观点之间提供了一种意想不到的联系。该奖项为将参与相关研究的研究生提供支持。更准确地说，Kashaev Volume猜想将纽结的有色Jones多项式的渐近性与其补的双曲体积联系起来。这个诱人的猜想现在已经有25年的历史了，并得到了许多启发式和实验证据的支持。然而，相应的性质只在极少数情况下得到了严格的证明。该项目的目标是在证明卡沙耶夫体积猜想的路线图中制定各种步骤，并特别强调其分析的微妙之处。作为第一步，该项目专注于圆上曲面丛的一个密切相关的猜想，其中量子不变量的组合与底层流形的几何更清楚地联系在一起。PI和他的合作者将在更特殊的情况下攻击这一猜想，以求在解决在此背景下出现的分析困难时非常具体。然后，PI将在第一个案例中收集的经验、见解和技术专业知识的基础上，一个接一个地继续路线图的下一步。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project is part of the general field of 3-dimensional topology and geometry, which analyzes 3-dimensional spaces with a high level of complexity and includes the study of the knotting phenomena that occur for curves in three dimensions. The underlying problems often arise from practical applications in the three-dimensional world we live in, but also from more theoretical considerations in mathematical physics. The project investigates a surprising connection between two very different approaches to 3-dimensional topology. The first approach is very algebraic and involves so-called quantum invariants for knotted curves in 3-dimensional spaces. The second approach is more geometric and analytic, and uses the non-euclidean geometry of the 3-dimensional hyperbolic space. The Kashaev Volume Conjecture, experimentally verified on many examples but still without a mathematical proof guaranteeing that it holds in all cases, provides an unexpected connection between these two viewpoints. The award provides support for graduate students who will be involved in related research.More precisely, the Kashaev Volume Conjecture connects the asymptotics of the colored Jones polynomial of a knot to the hyperbolic volume of its complement. This tantalizing conjecture is now 25-year old, and supported by much heuristic and experimental evidence. However, the corresponding property has been rigorously proved for only a very small number of cases. The goal of the project is to develop various steps in a roadmap towards a proof of the Kashaev Volume Conjecture, with a special emphasis on its analytic subtleties. As a first step, the project is focused on a closely related conjecture for surface bundles over the circle, where the combinatorics of quantum invariants are more clearly connected to the geometry of the underlying manifold. The PI and his collaborators will attack this conjecture in the even more special case of punctured torus bundles, in an effort to be very concrete while tackling the analytic difficulties that arise in this context. The PI will then build on the experience, insights and technical expertise gathered in this first case to proceed with the next steps in the roadmap, one after the other.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.