The Geometry and Topology of Quantum Invariants of Knots and 3-Manifolds

本项目涉及三维流形及其链环的量子不变量。一个主要的问题是，从拓扑学和几何学的角度来看，人们对这些不变量的理解很少。这个项目的目标是在量子不变量和现有的三维流形的几何和拓扑知识之间建立联系。主要方法是研究闭合三维流形的量子不变量的大水平渐近展开。研究这些渐近性的想法源于Witten关于量子不变量的工作。从这项工作和数学家随后的工作中，出现了一些关于这些渐近性的几何内容的严格的数学猜想。这些猜想表明量子不变量隐藏了深刻的拓扑信息。PI计划与其他人合作，通过具体的案例研究来核实(一些)猜测。例如，他将与J.E.Andersen共同继续他们在8字结上的手术项目。除了有限数量的三维流形外，所有这些流形都是双曲的，希望这个案例研究能够阐明量子不变量和双曲几何之间的联系。一个主要的问题是量子不变量是否能够检测双曲三维流形的体积。此外，Pi将尝试从更一般的角度工作，即他将应用T.Yoshida的新方法来计算所有闭合的3-流形的不变量的渐近展开，Yoshida的构造基于保形场论的简化。在这一点上，PI将与吉田合作，证明吉田的不变量与更成熟的Reshetikhin-Turaev不变量一致，从而与其他获得量子不变量的方法建立联系。这个项目试图开发新的技术来获得关于低维对象(如纽结和三维空间)的形状(拓扑)和几何的知识。我们应该研究低维空间及其几何和形状的一个主要原因是，自然界在许多情况下都包含这样的空间。因此，大分子，如DNA，具有“打结”的结构和新的研究方向，人们可以应用纽结不变量来获得关于这些分子的信息。它们的某些性质取决于它们的形状，例如它们是如何“打结”的。另一个例子是宇宙学，其中一个中心问题是：我们宇宙的“形状”是什么？在这里，人们应该想到这样一个事实：当我们在地球上走来走去时，它看起来就像一个普通的平面，但实际上地球表面就像一个非常大的球的表面。这与我们的宇宙相似。在局部上，一切看起来都是平的，也就是说，我们的宇宙在局部上看起来像一个标准的3维欧几里得空间(平面上的3维吊坠)，但也许整个宇宙是完全不同的东西，就像一个弯曲的紧凑空间。数学家使用所谓的不变量来检测空间的拓扑和几何性质，例如，它们是否像球面一样弯曲或不像平面一样弯曲。这个项目的主题是量子不变量，这是一个相对较新的不变量家族，可以追溯到V.Jones在80年代发现琼斯多项式。

This project concerns the quantum invariants of 3-manifolds and links in them. A main problem is that these invariants are poorly understood from the point of view of topology and geometry. The goal of this project is to make connections between the quantum invariants and the existing knowledge of the geometry and topology of 3-manifolds. The main approach will be an investigation of the large level asymptotic expansion of the quantum invariants of closed 3-manifolds. The idea of studying these asymptotics stems from Witten's work on the quantum invariants. From that work and subsequent work of mathematicians there have emerged some rigorous mathematical conjectures about the geometric content of these asymptotics. These conjectures indicate that the quantum invariants hide deep topological information. The PI plans to check, in collaboration with others, (some of) the conjectures via concrete case studies. E.g., he will jointly with J.E. Andersen continue their project on the surgeries on the figure 8 knot. All but a finite number of these 3-manifolds are hyperbolic and it is the hope that this case study can shed some light on the connection between quantum invariants and hyperbolic geometry. A main question is whether the quantum invariants can detect the volume of hyperbolic 3-manifolds. In addition the PI will try to work from a more general point of view, namely he will apply T. Yoshida's new approach to the quantum invariants to try to calculate the asymptotic expansion of the invariants for all closed 3-manifolds, Yoshida's construction being based on an abelianization of conformal field theory. In that connection the PI will jointly with Yoshida work on proving that Yoshida's invariants coincide with the more well established Reshetikhin--Turaev invariants thereby making a connection to the other approaches to the quantum invariants.This project tries to develop new techniques to obtain knowledge about the shape (topology) and geometry of low-dimensional objects such as knots and 3-dimensional spaces. A main reason that we should study low-dimensional spaces and their geometry and shape is that nature contains such spaces in many contexts. Thus large molecules, e.g. DNA, have a "knotted" structure and newer research points in the direction that one can apply knot invariants to obtain information about such molecules. Certain of their properties depend on their shape, e.g., how they are "knotted". Another example is in cosmology where a central question is: What is the "shape" of our universe? Here one should think of the fact that when we walk around on the earth it just looks like an ordinary plane but in fact the surface of the earth is like the surface of a very large ball. It is similar with our universe. Locally everything looks flat, i.e., locally our universe looks like a standard 3-dimensional Euclidean space (a 3-dimensional pendant to a plane), but maybe the universe as a whole is something completely different, like a curved compact space. Mathematicians use so-called invariants to detect topological and geometric properties of spaces, e.g., if they are curved like a sphere or not like a plane. The quantum invariants, the theme for this project, is a relatively new family of invariants dating back to V. Jones' discovery of the Jones polynomial in the 80ties.