瞬子Floer同调与Khovanov同调

瞬子Floer同调是利用Yang-Mills方程定义的三维流形以及三维流形中的链环的不变量。作为数学上第一个拓扑量子场论，瞬子Floer理论在低维拓扑中得到了广泛的应用。Khovanov同调是一个以纯组合的方式定义的链环不变量，被看作是是Jones多项式的范畴化。Khovanov同调和高能物理、辛拓扑、表示论等诸多领域都有着联系。最有趣的一点是，存在着联系Khovanov同调和Floer同调的谱序列。尽管定义的方式不复杂，但是Khovanov同调的几何本质依然没有被很好的理解。促进对其几何本质的理解的一个重要途径是证明Khovanov同调的侦测性结果。申请人计划进一步发展瞬子Floer同调，尤其是SU(N)结构群的高秩瞬子Floer同调，并用以证明纽结基本群的SU(N)表示的相关结果；同时利用瞬子Floer同调和Khovanov同调的联系来证明Khovanov同调对各类复杂链环的侦测性。

Instanton Floer homology is an invariant of 3-manifolds (or links in 3-manifolds) defined by Yang-Mills equations. As the first topological quantum field theory in mathematics, it has been used widely in the low dimensional topology. Khovanov homology is a link invariant defined in a purely combinatorial way. It can be regarded as a categorification of the Jones polynomial. Khovanov homology is related to high energy physics, symplectic topology and representation theory. Most interestingly, there exists spectral .sequences relating Khovanov homology to Floer theories. Even though the definition of Khovanov homology is not complicated, the geometric nature of Khovanov homology is still mysterious. One important way to understand the geometric nature of Khovanov homology is to study its detection property. The applicant plans to continue developing the instanton Floer theory, especially Instanton Floer theory of structure group SU(N), and use it to study the SU(N) representation of knot groups. Meanwhile, the applicant plans to study whether Khovanov homology detects specific classes of non-trivial links.