Spring School in Geometry and Quantum Topology

现代结理论始于琼斯多项式在八十年代中期的开创性发现。在Jones的发现之后不久，利用量子群理论和后来发展的TQFT构造了大量的多项式不变量。在这个领域的巨人中有菲尔兹奖获得者v.d rienfeld, m.k entsevich， V. Jones和E. Witten。虽然琼斯多项式是一个组合定义的对象，但它的某些极限包含了关于结补的代数拓扑和微分几何的重要信息。三个这样的限制是学校的重点：即体积猜想，AJ猜想和双曲几何。此外，Jones多项式的渐近性是可解拓扑量子场论的一个重要例子，它说明了弦论和m理论中的一些最新思想。几何极限还包含了三维物体的重要新算法，这也是该学派的主题之一。

Modern Knot Theory started with the seminal discovery of the Jones polynomial by V. Jones in the mid eighties. Shortly after Jones's discovery, a plethora of polynomial invariants were constructed using the theory of Quantum Groups, and the later developed TQFT. Among the giants in the field are Fields Medalists V. Drienfeld, M. Kontsevich, V. Jones and E. Witten. Although the Jones polynomial is a combinatorially defined object, certain limits of it contain important information about the algebraic topology and the differential geometry of the knot complement.Three such limits are the focus of the school: namely the Volume Conjecture, the AJ Conjecture and Hyperbolic Geometry. In addition, the asymptotics of the Jones polynomial is an important example of a solvable topological quantum field theory, which illustrates some of the latest ideas in string theory and M-theory. The geometric limits also contain nontrivial new arithmetic of 3-dimensional objects, which is also one of the themes of the school.