扭子理论已经成为研究多复变量及多四元变量理论的重要工具。本课题将在近年来研究工作的基础上，通过已经建立的复射影空间中某类开集上的一阶上同调类和四元空间中k-Cauchy-Fueter方程的解之间的一一对应，研究Penrose变换的逆变换，拟利用复射影空间中某些开集上的全纯函数和此开集上的(0,1)-形式之间的关系及Radon-Penrose变换的逆公式，给出确切的Penrose变换的逆公式；对已经建立的四元Heisenberg群上的切k-Cauchy–Fueter方程的解和某类上同调群之间的一一对应，寻找相应的积分公式实现此一一对应。这些问题的解决将丰富和完善Penrose 变换理论及四元k-Cauchy–Fueter算子理论，并将推动多四元变量理论的发展。

Twistor theory has become an important tool to investigate several complex and quaternionic variables theory. We intend to study the inverse of Penrose transformation by the correspondence between the first cohomology group over open subsets of the complex projective space and the solutions to k-Cauchy-Fueter equations on the quaternionic space,based on the recent work. We hope to give explicit inverse formula of Penrose transformation by the inverse formula of Radon-Penrose transformation and the relationship between the holomorphic functions on subsets of complex projective space and the (0,1)-forms on these subsets. We also want to find explicit integral formula to realize the one to one correspondence between some cohomology and the solutions to tangential k-Cauchy-Fueter equations on the quaternionic Heisenberg group. These work will enrich the theory of Penrose transformation and quaternionic k-Cauchy-Fueter operator and improve the development of theory of several quaternionic variables.