MATHEMATICAL PHYSICS,TOPOLOGY AND RELATED ALGEBRAIC STRUCTURE

1.基于共形场论的三维流形不变量的构造及其应用维滕在Chern-Simons规范理论的基础上提出了三维流形的拓扑不变量。后来从几何或组合的观点进行了几项工作。基于共形场论中映射类群的表示和3-流形的Heegaard分裂，构造了3-流形不变量.作为应用，利用共形场论中单值性的酉性，得到了Heegaard亏格和纽结隧道数等经典不变量的下界估计.研究了由Dynkin图自同构导出的对称性，我们精化了维滕不变量，建立了它的水平秩对偶。2.纽结空间上的图复形与微分形式本文研究的对象是无穷维纽结空间上的微分形式。我们在纽结空间上构造了一个从图复形到de Rham复形的态射。这可以被认为是环空间的杆复形的推广。特别地，作为图复形的零维上同调，Vassiliev不变量可以用Chern-Simons扰动理论中的积分来表示，将de Rham同伦理论应用于纯辫子群，证明了由Vassiliev不变量导出的纯辫子群的滤子与下中心级数一致.事实证明，Vassiliev不变量足够强大，可以区分任何纯辫子。

1.Construction of 3-manifold invariants derived from conformal field theory and its applicationsBased on Chern-Simons gauge theory, Witten proposed topological invariants of 3-manifolds. Several works have been done afterwards from geometric or combinatorial viewpoints. We constructed 3-manifold invariants based on representations of mapping class groups appearing in conformal field theory and Heegaard splitting of 3-manifolds.As an application, using the unitarity of the monodromy of conformal field theory, we obtained lower estimates for classical invariants, such as Heegaard genus and tunnel numbers of knots etc. Investigating the symmetry derived from Dynkin diagram automorphisms, we refined Witten invariant and established the level-rank duality.2.Graph complex and differential forms on knot spaceThe object of this research is differential forms on the space of all knots, which is an infinite dimensional space. We constructed a morphism from the graph complex to the de Rham complex on the knot space. This might be considered to be a generalization of the bar complex for the loop space. Especially, as the zero dimensional cohomology of the graph complex, the Vassiliev invariants can be represented by integrals appearing in Chern-Simons perturbation theory.Applying the de Rham homotopy theory to the pure braid group, we showed that the filtration derived from the Vassiliev invariants for pure braids coinsides with the lower central series. It turns out that the Vassiliev invariants are strong enough to distinguish any pure braid.