近年来，Gorenstein奇点理论成为非交换代数和非交换射影代数几何，特别是不变量理论中的核心研究内容。本项目从以下切入点研究有限群作用在Artin-Schelter正则代数上的不变子代数（即、非交换射影空间的轨道空间）的奇点问题：（1）引入分次Frobenius代数的Clifford形变以及非交换Milnor代数，从而得到非交换超曲面奇点代数（特别是非交换Klein奇点代数）的结构与表示的范畴刻画，特别地，给出低维超曲面奇点代数的等价分类；（2）构造Gorenstein不变子代数的非交换奇点解消，再通过奇异Hochschild上同调和Clifford超代数的表示范畴、以及AR对偶来刻画不同非交换奇点解消的联系。本项目的研究成果将为非交换射影空间的奇点解消和分类提供理论基础，也将为利用有限维代数研究Aritn-Schelter Gorenstein代数提供新的思路。

Recent years, the theory of Gorenstein singularities is one of the main research subjects in noncommutative algebra and noncommutative projective algebraic geometry, especially, in the invariant theory. In the present project, we mainly focus on the following questions to study the singularities of the invariant subalgebras of Artin-Schelter regular algebras under finite group actions (i.e., the orbit spaces of noncommutative projective spaces): (1) We will introduce the Clifford deformations of graded Frobenius algebras and noncommutative Milnor algebras, through which we obtain the categorical properties of the structures and representations of the noncommutative hypersurfaces, especially the noncommutative Klein singularities. In particularly, we will give a classification of the singularities of hypersurfaces with lower dimensions. (2) We will construct noncommutative quasi-resolutions of Gorenstein invariant subalgebras, and then establish the relations of different noncommutative quasi-resolutions through the singular Hochschild cohomology and the representation categories of corresponding Clifford superalgebras, and the Auslander-Reiten dualities. . The results of the present project will provide some theoretic preparations for constructing the resolutions of singularities, and to classify the singularities of noncommutative projective spaces. The project will provide a new method to study Artin-Schelter Gorenstein algebras by means of finite dimension algebras.