多项式代数的自同构和导子理论具有深刻的几何背景，并与组合理论、微分方程和各种代数结构理论不断产生交叉融合.. 本项目主要对多项式代数上的Keller映射和微分算子进行研究, 具体包括：(1) Keller映射的结构是多项式代数自同构理论和仿射代数几何的核心问题，我们将研究低维齐次(梯度)Keller映射的分类问题, 以及任意维低超越次数Keller映射的结构, 特别是其tame性；并研究wild自同构的(稳定)co-tame性; (2) 研究多项式代数上微分算子和导子的像, 利用Weyl代数的模理论研究高阶微分算子和局部幂零导子的像是否为Mathieu子空间的问题，并进一步研究Mathieu子空间的一般理论, 该问题源于Jacobi猜想; (3) 利用导子不变量理论以及同调代数和K-理论方法研究多项式代数的收缩的结构，该问题源于仿射簇的Zariski消去问题.

The theory of automorphisms and derivations of polynomial algebras has deep roots in geometry, and has continued crossing and integration with combination theory, differential equations and theories of several kinds of algebras.. In the project, we investigate automorphisms and differential operators of polynomial algebras including the following contents: (1) The structure of Keller maps is the central problem in the theory of polynomial automorphisms and affine algebraic geometry. We will study the classification problem of homogeneous (gradient) Keller maps in low dimensions and study the structure, especially the tameness, of Keller maps with low transcendence degree in arbitrary dimensions; Study the co-tameness of wild automorphisms. (2) Study the images of differential operators and derivations of polynomial algebras; use the module theory on Weyl algebras to study the problem of whether the images of higher-order differential operators and locally nilpotent derivations are Matheiu subspaces and study the general theory of Mathieu subspaces. This problem arose from the Jacobian conjecture. (3) Use the method of invariant theory of derivations, homological algebra and K-theory to study the structure of retracts of polynomial algebras. This problem arose from the Zariski cancellation problem for affine varieties.