非交换性是数学及其它领域中普遍存在且愈显重要的现象，而非交换代数是理解非交换性的数学基础；非交换代数几何中的新发展为非交换代数的研究提供了新的思想和方法；Artin-Schelter正则代数是几何思想运用于非交换代数研究的重要实现。本项目研究内容主要有两个方面：一是研究Artin-Schelter正则代数，其中整体维数为4的Artin-Schelter正则代数的分类问题是前一课题（批准号：10971188）的延续，将进而讨论它们的同调性质，另一关注的目标是5维Artin-Schelter正则代数所出现的新特性，以及从Hilbert序列获取正则代数的方法；二是利用EXT-代数上自然的A无穷结构给出同伦不变量，揭示一些非交换代数（如Koszul型代数，Calabi-Yau代数等）深层次信息，并描述它们的同调理论。对非交换代数理论更深入的理解将有助于数学（尤其是非交换代数几何）和物理的一些领域。

Noncommutativity is a very common and increasingly important phenomenon in mathematics and other disciplines. The subject of noncommutative algebras is the mathematical foundation for characterizing and understanding noncommutativity. Recent developments in noncommutative algebraic geometry furnish new ideas and methods for the study of noncommutative algebras. The study of Artin-Schelter regular algebras is the important realization of this new ideas. The proposal concerns mainly the following two topics in the area of noncommutative algebra: (1) Artin-Schelter regular algebras, one of goals is to classify the case, in which 4-dim is a continuous work of former project (grant no. 10971188); another goal we interested in is new phenomenon in the case of 5-dim and the method of gaining new Artin-Schelter regular algebras from the Hilbert series. (2) To give a homotopic invariant for connected graded algebras by using A-infinity algebra structures on the EXT-algebra, to reveal some deeper properties of noncommutative algebras such as Koszul-type algebras and Calabi-Yau algebras, and describe their homological theory. More understanding of noncommutativity will benefit to the study of mathematics (especially to noncommutative algebraic geometry) and some field of physics.