Calabi-Yau代数源于代数几何和数学物理中对Calabi-Yau流形的研究。Calabi-Yau代数和Calabi-Yau范畴因其在非交换代数及几何、代数表示论、共形场论、超弦理论中的深刻背景和广泛应用备受关注。该项目主要研究Calabi-Yau代数中的同调问题及相关课题。具体而言，研究Calabi-Yau代数的Yoneda Ext代数上的A_∞代数结构，及其分次稳定范畴的Calabi-Yau性质；计算分次Calabi-Yau代数的Hochschild同调与上同调,刻画其PBW形变的Calabi-Yau性质；研究Calabi-Yau代数在有限群作用下的不变子代数的结构与性质。以上研究，是Calabi-Yau代数同调理论的新探索，也是对非交换不变量理论的新尝试，将丰富和发展非交换代数及几何的理论。

Calabi-Yau algebras were originated in the research of Calabi-Yau manifolds in algebraic geometry and mathematical physics. Calabi-Yau algebras and Calabi-Yau categories attract lots of attention for their profound background and wide application in noncommutative algebraic geometry, algebra representation theory, conformed field theory, superstring theory. In this project, we mainly investigate the homology theory of Calabi-Yau algebras and some related topics. More explicitly, study the A_∞-algebra structure on the Yoneda Ext algebra of the Calabi-Yau algebra, and the Calabi-Yau property of its stable module category; and calculate the Hochschild homologies and cohomologies of Calabi-Yau algebras, then to characterize the Calabi-Yau property of its PBW deformation; also, we will explore the structure and property of the invariants of the Calabi-Yau algebra under the action of a finite group. The above research, is a new exploration on the homological theory of Calabi-Yau algebras, is also a new attempt on the noncommutative invariant theory and will enrich and develop the theory of noncommutative algebras and geometry.