本项目是代数表示论，代数几何，数学物理的交叉课题。我们主要研究代数与代数簇的导出等价，Hall 代数，丛倾斜代数，brane tiling 得到的 3-Calabi-Yau 代数等方面的问题。具体如下：（1）利用 2-周期复形范畴构造加权射影线上的半导出 Hall 代数，以及它与其它重要的 Hall 代数类，如加权射影线的 double Hall 代数，canonical 代数的半导出 Hall 代数以及 Bridgeland-Hall 代数的关系；（2）研究 weak Fano toric 曲面的倾斜 3-Calabi-Yau 代数的非交换 DT 不变量在 Seiberg 对偶，丛变换以及镜像对称下的关系；（3）证明有限型丛倾斜代数以及相关的粘合代数的 Cohen-Macaulay 模的稳定范畴可以实现为自入射代数的稳定范畴。

This proposal is about the intersection of representation theory of algebras, algebraic geometry and mathematical physics. It concerns derived equivalences between associative algebras and algebraic varieties, Hall algebras, cluster-tilted algebras and 3-Calabi-Yau algebras arising from brane tilings respectively. More precisely, (1) Use the category of 2-periodic complexes to construct the semi-derived Hall algebra of the weighted projective line and study the relations between this algebra and the double Hall algebra of the weighted projective line, the semi-derived Hall algebra and Bridgeland-Hall algebra of the corresponding canonical algebra; (2) Calculus the non-commutative Donaldson-Thomas (DT) invariants for the tilting 3-Calabi-Yau algebras of the weak Fano toric surfaces, and then get the wall-crossing formulas for the DT invariants under the Seiberg dualities, cluster mutations and mirror symmetry; (3) Prove that the stable categories of Cohen-Macaulay modules over the cluster-tilted algebras of Dynkin type and also the related gluing algebras can be realized by the stable categories of some self-injective algebras.