丛代数的组合与范畴化：方法与问题

研究(量子)丛代数，用范畴化方法和基于丛代数组合结构的组合方法。以问题解决带动理论和方法的创新。重点：.1. 量子丛代数的二次(多次)量子化问题.2. 丛代数的范畴化与高维丛代数的建立.3. wall-crossing序列、稳定函数条件、极大绿色序列等的关系.4. 用相容度函数刻画组合性质与系数变量的参数化.5. 可斜对称化量子丛代数的正性问题与量子双Bruhat包腔.6. F-多项式、丛变量的距离问题及极大绿色序列的最小长度。.预期目标：刻画基于Poisson结构的量子丛代数的多次量子化及其收敛性。用高维丛范畴，建立高维丛代数理论；争取完成完全符号斜对称丛代数及其正性的研究. 在三角范畴环境下完成极大绿色序列与稳定函数、散射图方法之间的关系。理解清楚相容度函数的本质；推动Y-系统的参数化研究。对更一般情况给出丛变量的Laurent展开式公式, 以此证明可斜对称化情形量子丛代数的正性猜想.

Study on (quantum) cluster algebras by means of categorification and combinatorial methods based on the combinatorial structure of cluster algebras. Promote the innovation of theories and methods via solving problems and conjectures. Our key points:..1. Second (multiple) quantization of quantum cluster algebras..2. The categorization of cluster algebras and the establishment of high dimensional cluster algebras..3. The relationship among wall-crossing sequences, stability functions/conditions, and maximal green sequences..4. Characterization of combinatorial properties by compatibility functions and parameterization of coefficient variables..5. The positivity conjecture of skew symmetrizable quantum cluster algebras and quantum double Bruhat cells..6. F-polynomials, the distance problem of cluster variables, and the minimum length of maximal green sequences...Expected goals: characterize multiple quantization and its convergence of quantum cluster algebras based on Poisson structures. By using high dimensional cluster categories, to establish the theory of high dimensional cluster algebras. Try to complete the study of totally sign-skew-symmetric cluster algebras and its positivity conjecture. Establish the relationship among maximal green sequences, stability functions and scattering diagrams in the setting of triangulated categories. To understand the nature of compatibility function and promote the parameterization of Y-systems. Give the Laurent expansion formula of cluster variables in more general case, and prove the positive conjecture of quantum cluster algebras in the skew-symmetrizable case.