在国际上几何化和范畴化大的数学研究潮流引导下，我们在已有的研究基础之上从几何方法入手围绕有限群及其表示理论中的核心热点问题展开研究，具体表现在两个方面的探讨上：量子同调代数与张量三角范畴。上述两个方向，最近几年来一直受到国际上来自不同领域的专家的共同关注，特别是在这一两年中逐渐成为相关领域研究的热点，有关的文章发表在一流的数学杂志上。在本项目中，我们将采取具体实例的探索和整体理论体系构建、完善相结合的研究路线，同时紧密关注上述理论在模表示论中，特别是模表示论三大支柱猜想中的应用。本项目的研究工作将要涉及到代数几何、微分几何等领域，体现了其多学科交叉与融合的特点。因此，本项目的成果也必将对上述相关领域的发展起到良好的推动作用。

To be in line with the international trends of geometrification and categorification, we will pursue our research based on what we have archieved by using geometric methods. There are two topics: Quantum Homological Algebra and Tensor Triangular Categories which are related to the hotspot themes in finite group and its representation theory. Despite of the long or short history, these themes have aroused the interest of many experts from different areas in resent years, and have become hot research topics in related fields. Related articles have published in first-class journal of mathematics. In this project, we will concern both concrete examples and improvement of theoretical framework in these topics. Forthermore, their applications in main conjuctures of modular representation are also highlighted. The study in this project will also involve the main ideology of algebraic geometry, differential geometry and other fields, which reflects its multi-disciplinary characteristics. Therefore, the results of this project will also play a good role in the related fields.