本项目研究齐性空间和纽结的的同调不变量，研究内容有以下两个方面：一、紧李群及Kac-Moody群及其齐性空间（特别是旗流形）的同调的计算。其中包括有理上同调和模p上同调及其上的Hopf代数结构的计算；广义上同调理论，主要是K-理论和配边理论；T-等变的广义上同调理论等等。二、纽结理论的同调不变量。研究主要集中在纽结对应的quandle和一般quandle的同调理论及其在低维拓扑中的应用。主要关心以下问题： 1. 纽结quandle到任一有限quandle同态的判定问题；2. quandle同调不变量的几何拓扑解释； 3. 正quandle同调群在纽结理论中的应用；4. 多个满足右分配律二元运算的quandle同调群的计算及其应用。上述研究都是与其它数学分支密切联系，引人关注的问题。其研究需要综合运用代数、几何、拓扑、表示论和组合等多方面的工具和方法，很有意义。

In this project we study the homological invariants of homogeneous spaces and knots.The research content has the following two aspects. The first is the cohomology of Kac-Moody groups and their homogeneous spaces(especially the flag manifolds). It contains: 1. The computation of rational cohomology, mop p cohomology and the corresponding Hopf algebra structure; 2. Generalized cohomology theory, in particularly the K - theory and the cobordism theory; 3. T-equivariant generalized cohomology theory and so on. The second is the cohomological invariants in Knot theory. The research focuses on the knot quandles and the cohomology theory of general quandles and its application in low dimensional topology. It contains: 1. The characterization of homomorphisms from knot quandle to finite quandles. 2. The topological interpretation of quandle cocycle invariants. 3. The applications of positive quandle homology in knot theory. 4. The calculations and applications of the homology of a multi-quandle with right distributive products.These problems are closely related to other branches of mathematics and are interesting. The research needs to integrate the tools and methods in algebra, geometry, topology, representation theory and combinatorics, so is very meaningful.